Find the absolute maximum and absolute minimum values of f(x)=2x^{2}-4\ln

Gregory Emery

Gregory Emery

Answered question

2021-12-07

Find the absolute maximum and absolute minimum values of f(x)=2x24ln(x) on the interval [12,2].

Answer & Explanation

poleglit3

poleglit3

Beginner2021-12-08Added 32 answers

Given
The given function is f(x)=2x24lnx
Theorem
If f(x) has absolute maximum at x=a, then f'(a)=0 and f''(a)
<0
If f(x) has absolute minimum at x=a, then f'(a)=0 and f''(a)
>0
solution
f(x)=2x24ln(x)
Diff. with respect to x,
f(x)=4x4x
put f'(x)=0
4x4x=0
4x24x=0
4x24=0
x21=0
x2=1
x=±1
x=1[12,1]
f(x) has absolute value at x=1
f(x)=4+4x2
f''(1)=8>0
f(x) has absolute minimum at x=1
Answer:
f(x) has absolute minimum at x=1

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