Question

How do you find the exact relative maximum and minimum of the polynomial function of `p(x) =80+108x-x^3`?

Answer 1

`"maximum at "(6,512)`
`"minimum at "(-6,-352)`

Explanation:

`"to determine turning points, differentiate and equate"`
`"to zero"`

`rArrp'(x)=108-3x^2`

`rArr3(36-x^2)=0`

`rArr3(x-6)(x+6)=0rArrx=+-6`

`p(6)=80+648-216=512`

`p(-6)=80-648+216=-352`

`"turning points at "(6,512)" and "(-6,-352)`

`"to determine the nature of the turning points"`

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

`• " if "p(x)>0" then minimum turning point"`

`• " if "p(x)<0" then maximum turning point"`

`p''(x)=-6x`

`p''(6)=-36<0" hence maximum"`

`p''(-6)=36>0" hence minimum"`

`"maximum at "(6,512)," minimum at "(-6,-352)`