How to find the critical numbers for f(x)=x^(-2)ln(x) to determine the maximum and minimum?

elhadaverdeibd7

elhadaverdeibd7

Answered question

2023-02-16

How to find the critical numbers for f ( x ) = x - 2 ln ( x ) to determine the maximum and minimum?

Answer & Explanation

kirungugd1v

kirungugd1v

Beginner2023-02-17Added 5 answers

Take the derivative of f ( x ) . You will need to use the product rule. You also need to know that the derivative of ln ( x ) is 1 x :
f ( x ) = x - 2 ( 1 x ) + ln ( x ) ( - 2 x - 3 )
f ( x ) = x - 3 - 2 ln ( x ) x - 3
Factor out a x - 3
x = 0 , e 1 2
Solve for x :
x = 0 , e 1 2
Plug in these numbers into the initial equation:
f ( 0 ) = 0 ln ( 0 ) = D N E .
We'll need to use limits for this:
lim x 0 x - 2 ln ( x ) = - . This is definitely a minimum
f ( e 1 2 ) = ( e - 1 ) ( ln ( e 1 2 ) ) = 1 2 e . This is a maximum because entering a value before or after it will produce one that is lower.

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