Find the absolute maximum and absolute minimum values of f on the given interval. f(x)=5+54x-2x^3, x in[0,4]

permaneceerc 2021-03-09 Answered
Find the absolute maximum and absolute minimum values of f on the given interval.
f(x)=5+54x2x3,x[0,4]
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Expert Answer

smallq9
Answered 2021-03-10 Author has 106 answers
Given,
f(x)=5+54x2x3,x[0,4]
Absolute maximum or absolute minimum values of a function on the given interval exists at a point at which its first derivative is zero or at the end points of the given interval.
So differentiating given function with respect to x, we get
f(x)=0+54(1)2(3x2)
f(x)=546x2
Now f'(x)=0 gives
546x2=0
6x2=54
x2=9
x=±3
But 3[0,4], therefore
x=3
Step 2
Now,
f(0)=5+54(0)2(0)3=5
f(3)=5+54(3)2(3)3=113
f(4)=5+54(4)2(4)3=93
Therefore absolute minimum value is 5 occurs at the point x = 0 and absolute maximum value is 113 occur at the point
x =3.
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