Find the points at which the function `f` given by `f(x)=(x-2)^4(x+1)^3` has local maxima and local minima and points of inflexion.

Explanation

The given function is `f(x)=(x-2)^{4}(x+1)^{3}.`


`therefore f^{prime}(x) =4(x-2)^{3}(x+1)^{3}+3(x+1)^{2}(x-2)^{4} `
`
=(x-2)^{3}(x+1)^{2}[4(x+1)+3(x-2)] `
`
=(x-2)^{3}(x+1)^{2}(7 x-2)`


Now, `f^{prime}(x)=0 Rightarrow x=-1 and x=frac{2}{7} or x=2`

Now, for values of x close to `frac{2}{7}` and to the of `frac{2}{7}, f^{prime}(x)>0`. Also, for values of x close to `frac{2}{7}` and to the of` frac{2}{7}, f^{prime}(x)<0.`

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