First Derivative Test - AP Calculus BC

1. What is the First Derivative Test used for?

   To determine relative (local) extrema of a function. Identifies local maxima and minima by analyzing sign changes of $f'(x)$.

2. What is the derivative of $f(x) = x^5 - 5x^3$?

   $f'(x) = 5x^4 - 15x^2$. Power rule applied to polynomial with multiple terms.

3. What test helps confirm if a critical point is an extremum?

   The First Derivative Test. Distinguishes between maxima, minima, and inflection points.

4. What is a critical point?

   Where $f'(x) = 0$ or $f'(x)$ is undefined. These are the only points where extrema can occur.

5. Evaluate $f'(x)$ at $x = 2$ for $f(x) = x^3 - 3x^2$.

   $f'(2) = 0$. Substitute $x = 2$ into $f'(x) = 3x^2 - 6x$.

6. State the condition for a critical point.

   $f'(x) = 0$ or $f'(x)$ is undefined. Necessary condition for potential extrema to exist.

7. What must be true for a critical point to be a local extremum?

   The derivative must change signs. Sign change of $f'(x)$ distinguishes extrema from inflection points.

8. What is the derivative of $f(x) = x^2 - 4x$?

   $f'(x) = 2x - 4$. Basic power rule: derivative of $x^2$ is $2x$.

9. Describe the behavior if $f'(x)$ changes from negative to positive.

   There's a relative minimum. Function reaches a valley; slope changes from downward to upward.

10. Determine extrema for $f(x) = x^2 - 4x + 3$ using the test.

    Minimum at $x = 2$. $f'(x) = 2x - 4 = 0$ at $x = 2$; sign changes from $-$ to $+$.

11. Determine if there is a maximum at $x = 0$ for $f(x) = -x^2$.

    Yes, maximum at $x = 0$. $f'(x) = -2x$ changes from positive to negative at origin.

12. What is a relative extremum?

    A point where $f(x)$ changes from increasing to decreasing or vice versa. Local maximum or minimum where function direction reverses.

13. Evaluate $f'(x)$ at $x = 0$ for $f(x) = \frac{1}{3}x^3 - x$.

    $f'(0) = -1$. Substitute $x = 0$ into $f'(x) = x^2 - 1$.

14. Find $f'(x)$: $f(x) = 2x^5 - 5x^4$.

    $f'(x) = 10x^4 - 20x^3$. Apply power rule: bring down exponent, reduce power by 1.

15. What is the next step after finding critical points in the test?

    Evaluate $f'(x)$ around critical points. Check sign changes of $f'(x)$ on intervals around each critical point.

16. Which test determines where a function is increasing or decreasing?

    The First Derivative Test. Sign of $f'(x)$ determines increasing/decreasing intervals.

17. What indicates a relative minimum using the First Derivative Test?

    Derivative changes from negative to positive. Sign change from $-$ to $+$ creates a valley in the graph.

18. Describe the behavior if $f'(x)$ changes from positive to negative.

    There's a relative maximum. Function reaches a peak; slope changes from upward to downward.

19. Identify the relative extrema: $f(x) = x^3 - 3x^2$.

    Relative maximum at $x = 0$; relative minimum at $x = 2$. $f'(x) = 3x^2 - 6x$; sign changes at $x = 0$ and $x = 2$.

20. What indicates a relative maximum using the First Derivative Test?

    Derivative changes from positive to negative. Sign change from $+$ to $-$ creates a peak in the graph.

21. What does $f'(x) > 0$ indicate about $f(x)$?

    $f(x)$ is increasing. Positive derivative means function values are rising.

22. Find the derivative: $f(x) = x^4 - 4x^3$.

    $f'(x) = 4x^3 - 12x^2$. Apply power rule: multiply by exponent, reduce exponent by 1.

23. Find $f'(x)$: $f(x) = \frac{x^3}{3} - x$.

    $f'(x) = x^2 - 1$. Standard power rule application with fractional coefficient.

24. What does $f'(x) = 0$ imply?

    Possible extremum; critical point. Potential location for maximum, minimum, or inflection point.

25. Find the derivative of $f(x) = x^3 - 3x^2 + 2x$.

    $f'(x) = 3x^2 - 6x + 2$. Apply power rule term by term to the polynomial.

26. Identify extrema for $f(x) = x^2 - 2x + 1$ using the test.

    Minimum at $x = 1$. $f'(x) = 2x - 2 = 0$ at $x = 1$; sign changes from $-$ to $+$.

27. What does $f'(x) < 0$ indicate about $f(x)$?

    $f(x)$ is decreasing. Negative derivative means function values are falling.

28. Determine relative extrema for $f(x) = x^3 - 3x + 1$ using the test.

    No relative extrema. $f'(x) = 3x^2 - 3 = 0$ at $x = \pm 1$; no sign changes.

29. Determine if there is an extremum at $x = 1$ for $f(x) = 2x^3 - 3x^2$.

    No extremum at $x = 1$. $f'(x) = 6x^2 - 6x$; no sign change at $x = 1$.

30. Determine relative extrema for $f(x) = x^2$ using the test.

    Minimum at $x = 0$. $f'(x) = 2x$; changes from $-$ to $+$ at $x = 0$.

31. Find $f'(x)$ for $f(x) = x^3 - 9x^2 + 27x$.

    $f'(x) = 3x^2 - 18x + 27$. Apply power rule to each term in the polynomial.

32. Determine if there is a minimum at $x = 0$ for $f(x) = x^2$.

    Yes, minimum at $x = 0$. $f'(x) = 2x$ changes from negative to positive at origin.

33. What is the derivative of $f(x) = \frac{1}{3}x^3 - x$?

    $f'(x) = x^2 - 1$. Apply power rule to each term separately.

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