Question

How many points in the interval [-a, a] does the instantaneous rate of change of f equal the average rate of change of f if `f(x)= e^(x^2)`?

Answer 1

At one point.

Explanation:

Average rate of change of `f` on `[-a,a]` is

`(f(a)-f(-a))/(a-(-a)) = (e^(a^2) - e^(a^2))/(2a) = 0`

The instantaneous rate of change of `f` at `x` is
`f'(x) = 2xe^(x^2)`

So the question becomes:

How many solutions to `2xe^(x^2)=0` are there in `[-a,a]`.

The answer is one (namely `x=0`).