Question

Let A(x)= int^ x f(t)dt, with f(x) shown in the graph below

Answer 1

Given:

`A(x)=\int_0^xf(t)dt`

The graph of f(x) is shown.

Let's find the following:

• (a). At what x value does A(x) have a local max?

The local maxima are the points where the function has a maximum value.

We have:

`\begin{gathered} A^(\prime)(x)=(d)/(dx)\int_0^xf(t)dt \\ \\ A^(\prime)(x)=f(x) \end{gathered}`

Using the graph, when f(x) =0, the values of x are:

x = 1, 3, 5

Which means the critical points are:

x = 1, 3, and 5

Now, let's apply the first derivative test which states that when A'(x) changes from negative to positive there is a local minimum at that point while when A'(x) changes from positive to negative, there is a local maximum at that point.

Using the graph, we have the following:

The change from negative to positive occur at: x = 1, 5. This means the local min occurs at these points.

The change from positive to negative occur at: x = 3. Hence, the local max occurs at this point.

ANSWER:

• Local max: x = 3

• Local min: x = 1, 5