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

First, we must find the second derivative of the function,

which was found using the following rules:

,  , ,

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

Using these values, we now create intervals on which to evaluate the sign of the second derivative:

Notice how at the bounds of the intervals, the second derivative is neither positive nor negative.

Evaluating the sign simply by plugging in any value on the given interval into the second derivative function, we find that on the first interval, the second derivative is negative, and on the second interval, the second derivative is positive. Thus, the function is concave down on the first interval, .

To determine the intervals on which the function is concave up, we must determine the intervals on which the function's second derivative is positive.

First, we must find the second derivative of the function,

which was found using the following rules:

, 

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

Using this value, we now create intervals on which to evaluate the sign of the second derivative:

Notice how at the bounds of the intervals, the second derivative is neither positive nor negative.

Evaluating the sign simply by plugging in any value on the given interval into the second derivative function, we find that on the first interval, the second derivative is negative, and on the second interval, the second derivative is positive. Thus, the function is concave up on the second interval, . 

To determine the intervals on which the function is concave up, we must determine the intervals on which the function's second derivative is positive.

First, we must find the second derivative of the function,

which was found using the following rules:

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

Using these values, we now create intervals on which to evaluate the sign of the second derivative:

Notice how at the bounds of the intervals, the second derivative is neither positive nor negative.

Evaluating the sign of the second derivative simply by plugging in any value on the given interval into the second derivative function, we find that on the first interval, the second derivative is positive, on the second interval, the second derivative is negative, and on the third interval, the second derivative is positive. Thus, the intervals on which the function is concave up are . 

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

First, we must find the second derivative of the function,

which was found using the following rules:

, 

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

Using these values, we now create intervals on which to evaluate the sign of the second derivative:

Notice how at the bounds of the intervals, the second derivative is neither positive nor negative.

Evaluating the sign simply by plugging in any value on the given interval into the second derivative function, we find that on the first interval, the second derivative is positive, on the second interval, the second derivative is negative, on the third interval, the second derivative is positive, on the fourth interval, the second derivative is negative, and on the fifth interval, the second derivative is positive. Thus, the function is concave down on . 

To determine the intervals on which the function is concave up, we must determine the intervals on which the function's second derivative is positive.

First, we must find the second derivative of the function,

which was found using the following rules:

, 

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

Using these values, we now create intervals on which to evaluate the sign of the second derivative:

Notice how at the bounds of the intervals, the second derivative is neither positive nor negative.

Evaluating the sign simply by plugging in any value on the given interval into the second derivative function, we find that on the first interval, the second derivative is positive, on the second interval, the second derivative is negative, and on the third interval, the second derivative is positive. Therefore, the function is concave up on . 

Explain whether f(x) is concave up or concave down when , and choose the correct explanation for its behavior.

To determine concavity, we need to find the second derivative of our function. Then, we will plug in the given value of x and see what the sign is. If f" is positive at the point, then we have a concave up curve. If it is negative, then we have concave down. If f" is zero at the point, then we have a point of inflection.

To find our derivatives, we need to recall two rules.

And

Using these two rules, we can find the derivative of f(x).

Our first term can be derived using our first rule. The derivative of e to the x is just e to the x.

This means that our first term will remain 16e to x.

For our other three terms, we follow the second rule. We will decrease each term's exponent by 1, and then multiply the coefficient by the old exponent.

Notice that the 13 will drop out. It is a constant term, and as such when we multiply it by it's original exponent (0) it wil be reduced to zero as well.

Clean up the above to get:

Using those same to rules, repeat the process to get our second derivative.

So with our second derivative, plug in 1 and find the sign.

We get a positive result, so our function is concave up.

So, we can say that f(x) is concave up, because 

To determine the intervals on which the function is concave up, we must determine the intervals on which the function's second derivative is positive.

First, we must find the second derivative of the function,

which was found using the following rules:

, 

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

Using these values, we now create intervals on which to evaluate the sign of the second derivative:

Notice how at the bounds of the intervals, the second derivative is neither positive nor negative.

Evaluating the sign simply by plugging in any value on the given interval into the second derivative function, we find that on the first interval, the second derivative is negative, on the second interval, the second derivative is positive, and on the third interval, the second derivative is negative. So, the function is concave up on .

We will find the points of inflection by setting our second derivative to zero.

Take the derivative again,

then set this equal to zero and reverse foil,

These are our points of inflection.

To find the inflection points of , we need to find  (which lucky for us, is already given!) set it equal to , and solve for .

. Start

. Divide by . We can do this, because  is never equal to .

On the unit circle, the values  cause , but only  is inside our interval . so  is the only value to consider here.

To prove that  is actually part of a point of inflection, we have to test an  value on the left and the right of , and substitute them into  and test their signs.

.

Hence , is the  coordinate of an inflection point of  on the interval .

Inflection points of a function occur when the second derivative equals zero. Therefore, we simply need to take two derivatives of our function, and solve.

Therefore, the two values that makes this function go to zero are