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Find the local maxima and local minima, if any, of the following functions. Find also the local maximum and the local minimum values, as the case may be `f(x)=x^{3}-6 x^{2}+9 x+15`

Text Solution

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`f(x)=x^{3}-6 x^{2}+9 x+15`

`therefore f^{prime}(x)=3 x^{2}-12 x+9`

`f^{prime}(x)=0 Rightarrow 3(x^{2}-4 x+3)=0 `

`Rightarrow 3(x-1)(x-3)=0`

`Rightarrow x=1,3`

`text { Now, } f^{circ} `

`(x)=6 x-12=6(x-2) `

`f^{prime prime}(1)=6(1-2)=-6<0 `

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