Question

f(x) = x^3-9x^2+10(A) List the a values of all local maxima of f.x values of local maximums =(B) List the a values of all local minima of f.x values of local minimums =(C) List the a values of all the inflection points of f. x values of inflection points :

Answer 1

Given function is,

A) To find values of all local maxima of f.

`f\mleft(x\mright)=x^3-9x^2+10`

Consider the derivative of f(x), we get,

`f\mleft(x\mright)=x^3-9x^2+10`
`f^(\prime)\left(x\right)`

we get,

`f^(\prime)\mleft(x\mright)=3x^2-18x^`

Also, let f'(x) be zero, we get (f'(x)=0),

`3x^2-18x=0`

Simplifing we get,

`3x\left(x-6\right)=0`

we get,

`x=0\text{ or x=6}`

To find the local maximum,

if f'(x-c)>0 anf f'(x+c)<0, then the x is local maximum

f f'(x-c)<0 anf f'(x+c)>0, then the x is local minimum

For x=0

Consider x-c as -1 (x-c=-1), we get

`f^(\prime)(-1)=3\left(-1\right)^2-18\left(-1\right)`

`f^(\prime)(-1)=21>0`

Consider x+c as 1, (x+c=1), we get

`f^(\prime)\mleft(1\mright)=3\left(1\right)^2-18\left(1\right)^`
`f^(\prime)\mleft(1\mright)=-15<0`

Since f'(-1)>0 anf f'(1)<0, then the 0 is local maximum.

x value of local maximum = 0

Answer is: x value of local maximum = 0