Local minimum and local maximum for calculus 1

Question

The function

This function has a local maximum at x=? with value?

This function has a local minimum at x=? with value?

Please show detailed solution on finding all values.

Answer 1

f(x) = 8x+9/x

f '(x) = 8-9/x²

Critical numbers are where f '(X) =0 or undefined

8-9/x²=0

8x²-9 =0

x= +sqrt(9/8) or -sqrt(9/8)

x= 1.06, -1.06

Using the first derivative test, f'(x) changes sign from negative to positive crossing x= 1.06, so x= 1.06 is a local min point and the min value is 16.97= f(1.06)

Similarly f '(x) changes sign from + to - crossing x= -1.06 , so x= -1.06 is a local max point and local max value is - 16.97

or x=0 ( here f'(x) is undefined. But x=0 is not in the domain of the function