Question

The function f(x)=x4−9x3+23x has a critical point at x=1. Use the second derivative test to identify it as a local maximum or local minimum. f(x) has a local maximum at x=1. f(x) has a local minimum at x=1.

Answer 1

x = 1 is a local maxima

Explanation:

Let us start by finding the first and second derivatives of this function

`f(x) = x^(4) -9x^(3)+23x\\f'(x) = 4x^(3)-27x^(2)  +23\\f''(x) = 12x^(2) -54x\\`

Now let's find the value of the second derivative at x=1.

`f''(1) = 12(1)^(2) -54(1) = -42`

So the second derivative is negative which means that there is a local maxima at x = 1