Find the absolute max and min values at the indicated interval f(x)=2x^3-x^2-4x+10 ,[-1,0]

Jerold 2021-02-12 Answered
Find the absolute max and min values at the indicated interval
f(x)=2x3x24x+10,[1,0]
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Expert Answer

Roosevelt Houghton
Answered 2021-02-13 Author has 106 answers
Given,
f(x)=2x3x24x+10,[1,0]
Absolute maximum and absolute minimum values of a function exist at a point where its first derivative is zero or at the end points of the given interval.
So first we find the derivative of the given function:
f(x)=23x22x4+0
f(x)=6x22x4
Now f'(x)=0
6x22x4=0
6x26x+4x4=0
6x(x1)+4(x1)=0
(x1)(6x+4)=0
x1=0or6x+4=0
x=1orx=23
Now computing the value of the function at these points and at the end points of the interval:
at x=1,f(1)=f(x)=2(1)3(1)24(1)+10=11
at x=0,f(0)=f(x)=2(0)3(0)24(0)+10=10
at x=23,f(23)=f(x)=2(23)3(23)24(23)+10=12.8148
We didn't find the value of f(1), because 1 does not belongs to the interval [1,0].
Hence absolute maximum value is 12.8148 and absolute minimum value is 10.
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