# Consider the function f(x)=2x^{3}+6x^{2}-90x+8, [-5,4] find the absolute minimum value of this function. find the absolute maximum value of this function.

Consider the function $$f(x)=2x^{3}+6x^{2}-90x+8, [-5,4]$$
find the absolute minimum value of this function.
find the absolute maximum value of this function.

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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.