First Derivative Test - AP Calculus AB

The derivative satisfies $k'(x)<0$ on $( -9,-4)$ and $k'(x)>0$ on $(-4,-1)$ . Where does $k$ have a local minimum?
Answer: At $x=-4$ , $k$ has a local minimum. This problem involves the First Derivative Test to identify local extrema based on the sign of the derivative. The First Derivative Test indicates that a local minimum occurs where the derivative changes from negative to positive. Here, k'(x) is negative on (-9,-4) and positive on (-4,-1), showing a sign change from negative to positive at x=-4. This means the function decreases before x=-4 and increases after, confirming a local minimum. A tempting distractor is choice B, which claims a local maximum at x=-4, but that would require a positive to negative change, not observed here. Remember, to apply this test effectively, always check for sign changes around critical points where the derivative is zero or undefined.
For $g$ on $[1,9]$ , $g'(x)>0$ on $(1,6)$ and $g'(x)<0$ on $(6,9)$ . Where does $g$ have a local maximum?
Answer: At $x=6$ , $g$ has a local maximum. This problem assesses the First Derivative Test. The First Derivative Test determines local extrema by examining sign changes in the first derivative around critical points. If f' changes from positive to negative at a point, it indicates a local maximum there. If it changes from negative to positive, it indicates a local minimum, and no sign change means no extremum. In this case, at x=6, g' changes from positive on (1,6) to negative on (6,9), confirming a local maximum. A tempting distractor is choice A, which claims a local minimum at x=6, but this fails because the sign change is from positive to negative, not the reverse. To apply this generally, always identify points where the derivative might be zero and check the signs immediately to the left and right for extrema.
For $p$ , $p'(x)>0$ on $(-2,0)$ and $p'(x)>0$ on $(0,5)$ . What happens at $x=0$ ?
Answer: At $x=0$ , $p$ has no local extremum. This problem involves the First Derivative Test to identify local extrema based on the sign of the derivative. The First Derivative Test requires a sign change in the derivative to indicate an extremum; no change means none. Here, p'(x) is positive on (-2,0) and positive on (0,5), showing no sign change at x=0. Thus, the function is increasing on both sides, so no local extremum at x=0. A tempting distractor is choice B, suggesting a local minimum at x=0, but without a sign change from negative to positive, this is incorrect. Remember, to apply this test effectively, always check for sign changes around critical points where the derivative is zero or undefined.
A function $t$ has $t'(x)<0$ for $x< -1$ and $t'(x)>0$ for $x>-1$ . What occurs at $x=-1$ ?
Answer: At $x=-1$ , $t$ has a local minimum. This problem involves the First Derivative Test to identify local extrema based on the sign of the derivative. The First Derivative Test indicates that a local minimum occurs where the derivative changes from negative to positive. Here, t'(x) is negative for x<-1 and positive for x>-1, showing a sign change from negative to positive at x=-1. This means the function decreases before x=-1 and increases after, confirming a local minimum. A tempting distractor is choice A, which claims a local maximum at x=-1, but that would require a positive to negative change, not observed here. Remember, to apply this test effectively, always check for sign changes around critical points where the derivative is zero or undefined.
A function $T$ has $T'(x)<0$ on $( -6,-1)$ and $T'(x)>0$ on $(-1,0)$ . Where does $T$ have a local minimum?
Answer: At $x=-1$ , $T$ has a local minimum. This problem assesses the First Derivative Test. The First Derivative Test determines local extrema by examining sign changes in the first derivative around critical points. If f' changes from negative to positive at a point, it indicates a local minimum there. If it changes from positive to negative, it indicates a local maximum, and no sign change means no extremum. In this case, at x=-1, T' changes from negative on (-6,-1) to positive on (-1,0), confirming a local minimum. A tempting distractor is choice D, which claims a local maximum at x=-1, but this fails because the sign change is from negative to positive, not positive to negative. To apply this generally, always identify points where the derivative might be zero and check the signs immediately to the left and right for extrema.
The derivative satisfies $A'(x)<0$ on $( -2,3)$ and $A'(x)>0$ on $(3,7)$ . Where does $A$ have a local minimum?
Answer: At $x=3$ , $A$ has a local minimum. This problem assesses the First Derivative Test. The First Derivative Test determines local extrema by examining sign changes in the first derivative around critical points. If f' changes from negative to positive at a point, it indicates a local minimum there. If it changes from positive to negative, it indicates a local maximum, and no sign change means no extremum. In this case, at x=3, A' changes from negative on (-2,3) to positive on (3,7), confirming a local minimum. A tempting distractor is choice B, which claims a local maximum at x=3, but this fails because the sign change is from negative to positive, not positive to negative. To apply this generally, always identify points where the derivative might be zero and check the signs immediately to the left and right for extrema.
A function $S$ has $S'(x)>0$ on $( -6,-1)$ and $S'(x)<0$ on $(-1,0)$ . Where does $S$ have a local maximum?
Answer: At $x=-1$ , $S$ has a local maximum. This problem assesses the First Derivative Test. The First Derivative Test determines local extrema by examining sign changes in the first derivative around critical points. If f' changes from positive to negative at a point, it indicates a local maximum there. If it changes from negative to positive, it indicates a local minimum, and no sign change means no extremum. In this case, at x=-1, S' changes from positive on (-6,-1) to negative on (-1,0), confirming a local maximum. A tempting distractor is choice A, which claims a local minimum at x=-1, but this fails because the sign change is from positive to negative, not the reverse. To apply this generally, always identify points where the derivative might be zero and check the signs immediately to the left and right for extrema.
A differentiable function $H$ has $H'(x)<0$ on $( -9,-7)$ and $H'(x)>0$ on $(-7,-3)$ . Where does $H$ have a local minimum?
Answer: At $x=-7$ , $H$ has a local minimum. This problem assesses the First Derivative Test. The First Derivative Test determines local extrema by examining sign changes in the first derivative around critical points. If f' changes from negative to positive at a point, it indicates a local minimum there. If it changes from positive to negative, it indicates a local maximum, and no sign change means no extremum. In this case, at x=-7, H' changes from negative on (-9,-7) to positive on (-7,-3), confirming a local minimum. A tempting distractor is choice B, which claims a local maximum at x=-7, but this fails because the sign change is from negative to positive, not positive to negative. To apply this generally, always identify points where the derivative might be zero and check the signs immediately to the left and right for extrema.
The derivative satisfies $B'(x)>0$ on $( -2,3)$ and $B'(x)<0$ on $(3,7)$ . Where does $B$ have a local maximum?
Answer: At $x=3$ , $B$ has a local maximum. This problem assesses the First Derivative Test. The First Derivative Test determines local extrema by examining sign changes in the first derivative around critical points. If f' changes from positive to negative at a point, it indicates a local maximum there. If it changes from negative to positive, it indicates a local minimum, and no sign change means no extremum. In this case, at x=3, B' changes from positive on (-2,3) to negative on (3,7), confirming a local maximum. A tempting distractor is choice C, which claims a local minimum at x=3, but this fails because the sign change is from positive to negative, not the reverse. To apply this generally, always identify points where the derivative might be zero and check the signs immediately to the left and right for extrema.
For $q$ , $q'(x)>0$ on $(-6,-2)$ and $q'(x)<0$ on $(-2,1)$ . Where does $q$ have a local maximum?
Answer: At $x=-2$ , $q$ has a local maximum. This problem involves the First Derivative Test to identify local extrema based on the sign of the derivative. The First Derivative Test states that a local maximum occurs where the derivative changes from positive to negative. Here, q'(x) is positive on (-6,-2) and negative on (-2,1), showing a sign change from positive to negative at x=-2. This means the function increases before x=-2 and decreases after, confirming a local maximum. A tempting distractor is choice D, which claims a local minimum at x=-2, but that would require a negative to positive change, not observed here. Remember, to apply this test effectively, always check for sign changes around critical points where the derivative is zero or undefined.
A differentiable function $r$ has $r'(x)<0$ on $(1,5)$ and $r'(x)>0$ on $(5,9)$ . Where is a local minimum?
Answer: At $x=5$ , $r$ has a local minimum. This problem involves the First Derivative Test to identify local extrema based on the sign of the derivative. The First Derivative Test indicates that a local minimum occurs where the derivative changes from negative to positive. Here, r'(x) is negative on (1,5) and positive on (5,9), showing a sign change from negative to positive at x=5. This means the function decreases before x=5 and increases after, confirming a local minimum. A tempting distractor is choice C, which claims a local maximum at x=5, but that would require a positive to negative change, not observed here. Remember, to apply this test effectively, always check for sign changes around critical points where the derivative is zero or undefined.
The derivative $W'(x)$ is negative on $(0,4)$ and positive on $(4,12)$ . Where does $W$ have a local minimum?
Answer: At $x=4$ , $W$ has a local minimum. This problem assesses the First Derivative Test. The First Derivative Test determines local extrema by examining sign changes in the first derivative around critical points. If f' changes from negative to positive at a point, it indicates a local minimum there. If it changes from positive to negative, it indicates a local maximum, and no sign change means no extremum. In this case, at x=4, W' changes from negative on (0,4) to positive on (4,12), confirming a local minimum. A tempting distractor is choice A, which claims a local maximum at x=4, but this fails because the sign change is from negative to positive, not positive to negative. To apply this generally, always identify points where the derivative might be zero and check the signs immediately to the left and right for extrema.
On $(-8,-3)$ , $f'(x)>0$ ; on $(-3,2)$ , $f'(x)<0$ ; on $(2,6)$ , $f'(x)>0$ . Where is a local maximum?
Answer: At $x=-3$ , $f$ has a local maximum. This problem involves the First Derivative Test to identify local extrema based on the sign of the derivative. The First Derivative Test states that a local maximum occurs where the derivative changes from positive to negative. Here, f'(x) >0 on (-8,-3) and <0 on (-3,2), showing a + to - change at x=-3; then <0 to >0 at x=2, which is a min. Thus, the local maximum is at x=-3. A tempting distractor is choice A, suggesting a max at x=2, but that's actually a min due to - to + change. Remember, to apply this test effectively, always check for sign changes around critical points where the derivative is zero or undefined.
For $Q$ , $Q'(x)<0$ on $( -3,-2)$ and $Q'(x)>0$ on $(-2,-1)$ . Where does $Q$ have a local minimum?
Answer: At $x=-2$ , $Q$ has a local minimum. This problem assesses the First Derivative Test. The First Derivative Test determines local extrema by examining sign changes in the first derivative around critical points. If f' changes from negative to positive at a point, it indicates a local minimum there. If it changes from positive to negative, it indicates a local maximum, and no sign change means no extremum. In this case, at x=-2, Q' changes from negative on (-3,-2) to positive on (-2,-1), confirming a local minimum. A tempting distractor is choice A, which claims a local maximum at x=-2, but this fails because the sign change is from negative to positive, not positive to negative. To apply this generally, always identify points where the derivative might be zero and check the signs immediately to the left and right for extrema.
The derivative satisfies $h'(x)<0$ on $(-5,-1)$ and $h'(x)<0$ on $(-1,3)$ . What local extremum occurs at $x=-1$ ?
Answer: At $x=-1$ , $h$ has no local extremum. This problem involves the First Derivative Test to identify local extrema based on the sign of the derivative. The First Derivative Test requires a sign change in the derivative to indicate an extremum; no change means none. Here, h'(x) is negative on (-5,-1) and negative on (-1,3), showing no sign change at x=-1. Thus, the function is decreasing on both sides, so no local extremum at x=-1. A tempting distractor is choice B, suggesting a local minimum at x=-1, but without a sign change from negative to positive, this is incorrect. Remember, to apply this test effectively, always check for sign changes around critical points where the derivative is zero or undefined.
A function $m$ has $m'(x)>0$ on $( -1,4)$ and $m'(x)<0$ on $(4,10)$ . Where does $m$ have a local maximum?
Answer: At $x=4$ , $m$ has a local maximum. This problem involves the First Derivative Test to identify local extrema based on the sign of the derivative. The First Derivative Test states that a local maximum occurs where the derivative changes from positive to negative. Here, m'(x) is positive on (-1,4) and negative on (4,10), showing a sign change from positive to negative at x=4. This means the function increases before x=4 and decreases after, confirming a local maximum. A tempting distractor is choice A, which claims a local minimum at x=4, but that would require a negative to positive change, not observed here. Remember, to apply this test effectively, always check for sign changes around critical points where the derivative is zero or undefined.
For $f$ on $[1,9]$ , $f'(x)<0$ on $(1,6)$ and $f'(x)>0$ on $(6,9)$ . Where does $f$ have a local minimum?
Answer: At $x=6$ , $f$ has a local minimum. This problem assesses the First Derivative Test. The First Derivative Test determines local extrema by examining sign changes in the first derivative around critical points. If f' changes from negative to positive at a point, it indicates a local minimum there. If it changes from positive to negative, it indicates a local maximum, and no sign change means no extremum. In this case, at x=6, f' changes from negative on (1,6) to positive on (6,9), confirming a local minimum. A tempting distractor is choice C, which claims a local maximum at x=6, but this fails because the sign change is from negative to positive, not positive to negative. To apply this generally, always identify points where the derivative might be zero and check the signs immediately to the left and right for extrema.
A function $P$ satisfies $P'(x)>0$ on $( -2,2)$ and $P'(x)<0$ on $(2,2.5)$ . Where does $P$ have a local maximum?
Answer: At $x=2$ , $P$ has a local maximum. This problem assesses the First Derivative Test. The First Derivative Test determines local extrema by examining sign changes in the first derivative around critical points. If f' changes from positive to negative at a point, it indicates a local maximum there. If it changes from negative to positive, it indicates a local minimum, and no sign change means no extremum. In this case, at x=2, P' changes from positive on (-2,2) to negative on (2,2.5), confirming a local maximum. A tempting distractor is choice D, which claims a local minimum at x=2, but this fails because the sign change is from positive to negative, not the reverse. To apply this generally, always identify points where the derivative might be zero and check the signs immediately to the left and right for extrema.
A function $u$ has $u'(x)>0$ on $( -3,0)$ and $u'(x)<0$ on $(0,2)$ . What occurs at $x=0$ ?
Answer: At $x=0$ , $u$ has a local maximum. This problem involves the First Derivative Test to identify local extrema based on the sign of the derivative. The First Derivative Test states that a local maximum occurs where the derivative changes from positive to negative. Here, u'(x) is positive on (-3,0) and negative on (0,2), showing a sign change from positive to negative at x=0. This means the function increases before x=0 and decreases after, confirming a local maximum. A tempting distractor is choice A, which claims a local minimum at x=0, but that would require a negative to positive change, not observed here. Remember, to apply this test effectively, always check for sign changes around critical points where the derivative is zero or undefined.
For $w$ on $[-2,5]$ , $w'(x)>0$ on $(-2,1)$ and $w'(x)>0$ on $(1,5)$ . What happens at $x=1$ ?
Answer: At $x=1$ , $w$ has no local extremum. This problem involves the First Derivative Test to identify local extrema based on the sign of the derivative. The First Derivative Test requires a sign change in the derivative to indicate an extremum; no change means none. Here, w'(x) is positive on (-2,1) and positive on (1,5), showing no sign change at x=1. Thus, the function is increasing on both sides, so no local extremum at x=1. A tempting distractor is choice A, suggesting a local maximum at x=1, but without a sign change from positive to negative, this is incorrect. Remember, to apply this test effectively, always check for sign changes around critical points where the derivative is zero or undefined.

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