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

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

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Mayme
Step 1
The function is given by,
$$\displaystyle{f{{\left({x}\right)}}}={2}{x}^{{{3}}}+{6}{x}^{{{2}}}-{90}{x}+{8}$$
Step 2
Differentiatte with respect to x,
$$\displaystyle{f}'{\left({x}\right)}={6}{x}^{{{2}}}+{12}{x}-{90}$$
Solve f'(x)=0.
f'(x)=0
$$\displaystyle{6}{x}^{{{2}}}+{12}{x}-{90}={0}$$
$$\displaystyle{x}^{{{2}}}+{2}{x}-{15}={0}$$
(x+5)(x-3)=0
x=-5,3
Step 3
Find the functional value at x=–5.
$$\displaystyle{f{{\left(-{5}\right)}}}={2}{\left(-{5}\right)}^{{{3}}}+{6}{\left(-{5}\right)}^{{{2}}}-{90}{\left(-{5}\right)}+{8}$$
=-250+150+450+8
=358
Find the functional value at x=3.
$$\displaystyle{f{{\left({3}\right)}}}={2}{\left({3}\right)}^{{{3}}}+{6}{\left({3}\right)}^{{{2}}}-{90}{\left({3}\right)}+{8}$$
=54+36-270+8
=-172
The absolute minimum of the given function is -172.
The absolute maximum of the given function is 358.