Question

# Find the absolute maximum and minimum values of f on the given interval. f(x)=4x^{3}-6x^{2}-24x+9. [-2,3]

Functions
Find the absolute maximum and minimum values of f on the given interval.
$$f(x)=4x^{3}-6x^{2}-24x+9. [-2,3]$$

2021-05-10
Step 1
We will find the first derivative using the power rule
$$f(x)=4x^{3}-6x^{2}-24x+9$$
$$f'(x)=4(3x^{2})-6(2x)-24$$
$$f'(x)=12x^{2}-12x-24$$
Step 2
We find the critical values by solving f'(x)=0
$$12x^{2}-12x-24=0$$
$$12(x^{2}-x-2)=0$$
12(x-2)(x+1)=0
x=2,-1
Then we find the values of the function at the critical points and at the endpoints.
$$f(x)=4x^{3}-6x^{2}-24x+9$$
$$f(-2)=4(-2)^{3}-6(-2)^{2}-24(-2)+9=1$$
$$f(-1)=4(-1)^{3}-6(-1)^{2}-24(-1)+9=23$$ (max)
$$f(2)=4(2)^{3}-6(2)^{2}-24(2)+9=-31$$ (min)
$$f(3)=4(3)^{3}-6(3)^{2}-24(3)+9=-9$$