Question

f(x) = x³ $-$ 3x

Answer 1

$$\begin{gathered} \mathrm{Given}:\hspace{0.17em}f\left(x\right)=x^3-3x \\ \Rightarrow f\text{'}\left(x\right)=3x^2-3 \\ \mathrm{For}\mathrm{a}\mathrm{local}\mathrm{maximum}\mathrm{or}\mathrm{a}\mathrm{local}\mathrm{minimum},\mathrm{we}\mathrm{must}\mathrm{have} \\ f\text{'}\left(x\right)=0 \\ \Rightarrow 3x^2-3=0 \\ \Rightarrow x^2-1=0 \\ \Rightarrow x=\pm 1 \end{gathered}$$



Since f '(x) changes from negative to positive as x increases through 1, x = 1 is the point of local minima.
The local minimum value of f (x) at x = 1 is given by
${\left(1\right)}^3-3\left(1\right)=-2$

Since f '(x) changes from positive to negative when x increases through -1, x = -1 is the point of local maxima.
The local maximum value of f (x) at x = -1 is given by
${\left(-1\right)}^3-3\left(-1\right)=2$



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