Step 1

The function is given by,

\(f(x)=2x^{3}+6x^{2}-90x+8\)

Step 2

Differentiatte with respect to x,

\(f'(x)=6x^{2}+12x-90\)

Solve \(f'(x)=0\).

\(f'(x)=0\)

\(6x^{2}+12x-90=0\)

\(x^{2}+2x-15=0\)

\((x+5)(x-3)=0\)

\(x=-5,3\)

Step 3

Find the functional value at \(x=–5.\)

\(f(-5)=2(-5)^{3}+6(-5)^{2}-90(-5)+8\)

\(=-250+150+450+8\)

\(=358\)

Find the functional value at \(x=3\).

\(f(3)=2(3)^{3}+6(3)^{2}-90(3)+8\)

\(=54+36-270+8\)

\(=-172\)

The absolute minimum of the given function is -172.

The absolute maximum of the given function is 358.