To find the intervals on which the function is concave down, we must find the intervals on which the second derivative of the function is negative.

First, we must find the first and second derivatives:

The derivatives were found using the following rule:

Next, we must find the values at which the second derivative is equal to zero:

Now, we can make the intervals:

Note that at the bounds of the intervals the second derivative is neither positive nor negative.

To determine the sign of the second derivative on the intervals, simply plug in any value on the interval into the second derivative function; on the first interval, the second derivative is positive, on the second it is negative, and on the third it is positive. Thus, the function is concave down on the interval .

To find the intervals on which the function is decreasing, we must find the intervals on which the first derivative of the function is negative.

So, the first derivative of the function is equal to

and was found using the following rule:

Next, we must find the critical values, at which the first derivative is equal to zero:

Now, we make the intervals, using c as our upper and lower bound:

To determine whether the first derivative is positive or negative each interval, simply plug in any number on the interval into the first derivative function. On the first interval, the first derivative is positive, while on the second interval, the first derivative is negative. Our answer is therefore .

To find when a function is concave, you must first take the 2nd derivative, then set it equal to 0, and then find between which zero values the function is negative.

First, find the 2nd derivative:

Set equal to 0 and solve:

Now test values on all sides of these to find when the function is negative, and therefore decreasing. I will test the values of -3 and 0.

Since the value that is negative is when x=-3, the interval is decreasing on the interval that includes 0. Therefore, our answer is:

Is the function g(x) increasing or decreasing on the interval ?

To tell if a function is increasing or decreasing, we need to see if its first derivative is positive or negative. let's find g'(x)

Recall that the derivative of a polynomial can be found by taking each term, multiplying by its exponent and then decreasing the exponent by 1.

Next, we need to see if the function is positive or negative over the given interval.

Begin by finding g'(-5)

So, g'(-5) is negative, but what about g'(0)?

Also negative, so our answer is:

Decreasing, because g'(x) is negative on the interval .

The function is decreasing when the first derivative is negative. We first find when the derivative is zero. To find the derivative, we apply the quotient rule,

.

Therefore the derivative is zero at . To find when it is negative plug in test points on each of the three intervals created by these zeros.

For instance,

.

Hence the function is decreasing on

.

To determine the intervals on which the function is concave down, we must find the intervals on which the second derivative of the function is negative.

First, we must find the second derivative:

The derivatives were found using the following rule:

Now, we must find the value at which the second derivative is equal to zero.

We will now use this as the upper and lower limit of our intervals on which we evaluate the sign of the second derivative:

On the first interval, the second derivative is negative, while on the second interval, the second derivative is positive. Thus, our answer is .

This parabola would have the formula . When the first derivative is positive, the function is parabola is increasing. The first derivative is , which is positive on the domain . When the second derivative is positive, the function is concave up. The second derivative is , which is always positive for all real values of .

Therefore, this function is,

Concave up over and increasing over .

Tell whether f(x) is concave up or concave down on the interval \[1,2\]

To find concave up and concave down, we need to find the second derivative of f(x).

Let's begin by finding f'(x)

Next find f ''(x)

Now, to test for concavity, plug in the endpoints of the interval:

So, on this interval, f"(x) will always be negative. This means that our function is concave down on this interval.

To test concavity, we must first find the second derivative of f(x)

This function is concave down anywhere that f''(x)<0, so...

So,

for all

So on the interval -5,-4 f(x) is concave down because f''(x) is negative.

Is the function b(t) concave up, concave down, or neither when t is equal to -3?

To test for concavity, we need to find the sign of the function's second derivative at the given time.

Begin by recalling that the derivative of a polynomial is found by multiplying each term by its exponent, then decreasing the exponent by 1.

Doing this gets us the following:

Almost there, but we need b"(-3)

b"(-3) is negative, therefore our function is concave down when t=-3