Question

What are the points of inflection, if any, of `f(x)=2x^4-e^(8x`?

Answer 1

See below

Explanation:

First step is finding the second derivative of the function

`f(x)=2x^4-e^(8x)`

`f'(x)=8x^3-8e^(8x)`

`f''(x)=24x^2-64e^(8x)`

Then we must find a value of x where:

`f''(x)=0`

(I used a calculator to solve this)

`x=-0.3706965`

So at the given `x`-value, the second derivative is 0. However, in order for it to be a point of inflection, there must be a sign change around this `x` value.

Hence we can plug values into the function and see what happens:

`f(-1)=24-64e^(-8)` definetly positive as `64e^(-8)` is very small.

`f(1)=24-64e^(8)` definetly negative as `64e^8` is very big.

So there is a sign change around `x=-0.3706965`, so it is therefore an inflection point.