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 :

Answer:

Given function is,

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

[tex]f\mleft(x\mright)=x^3-9x^2+10[/tex]

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

[tex]f\mleft(x\mright)=x^3-9x^2+10[/tex][tex]f^{\prime}\left(x\right)[/tex]

we get,

[tex]f^{\prime}\mleft(x\mright)=3x^2-18x^[/tex]

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

[tex]3x^2-18x=0[/tex]

Simplifing we get,

[tex]3x\left(x-6\right)=0[/tex]

we get,

[tex]x=0\text{ or x=6}[/tex]

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

[tex]f^{\prime}(-1)=3\left(-1\right)^2-18\left(-1\right)[/tex]

[tex]f^{\prime}(-1)=21>0[/tex]

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

[tex]f^{\prime}\mleft(1\mright)=3\left(1\right)^2-18\left(1\right)^[/tex][tex]f^{\prime}\mleft(1\mright)=-15<0[/tex]

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

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