Question

How do you find the local maximum and minimum values of `f(x) = 7x + 9x^(-1)`?

Answer 1

`"local min at " ((3sqrt7)/7,6sqrt7)`

`"local max at " (-(3sqrt7)/7,-6sqrt7)`

Explanation:

`"to find the x-coordinates of the stationary points, differentiate"`
`f(x)" and equate to zero"`

`f'(x)=7-9x^-2=0`

`rArr9/x^2=7`

`rArrx^2=9/7rArrx=+-3/sqrt7=+-(3sqrt7)/7`

`f((3sqrt7)/7)=7((3sqrt7)/7)+9/((3sqrt7)/7)=6sqrt7`

`f(-(3sqrt7)/7)=-6sqrt7`

`rArr"stationary points at " ((3sqrt7)/7,6sqrt7)" and"`

`(-(3sqrt7)/7,-6sqrt7)`

`"to find the nature of the stationary points"`

`"use the "color(red)"second derivative test"`

`f'(x)=7-9x^-2`

`rArrf''(x)=18x^-3=18/x^3`

`f''((3sqrt7)/7)>0rArrcolor(red)" local minimum"`

`f''(-(3sqrt7)/7)<0rArrcolor(red)" local maximum"`

`rArr((3sqrt7)/7,6sqrt7)" is a local minimum"`

`"and " (-(3sqrt7)/7,-6sqrt7)" is a local maximum"` graph{7x+9x^-1 [-38.9, 38.88, -19.45, 19.45]}