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Find the extrema points of f(x)=3x^(4)...

Find the extrema points of ` f(x)=3x^(4)-4x^(3)-36x^(2)+28`

Text Solution

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`f'(x)=12x^(3)-12x^(2)-72x `
`=12x(x+2)(x-3)`
Using first order derivative test ,
We determine the sign of `f'(x)` using a number line .

We observe that
` x=-2 , 3 ` (Points of local minima )
x=0 (Points of local maxima )
Using seond order derivative test
`f'(x)=12x(x+2)(x-3)`
`f''(x)=36x^(2)-24x-72`
` =12(3x^(2)-2x-6)`
`f''(0) lt 0implies ` x=0 is a local maximum
`f''(-2) gt 0 implies` x=-2 is a local minimum
`f''(3) gt 0 implies ` x=3 is a local minimum
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