Question

Find local maxima and local minima for the function f(x) $=x^3-3x$

• A. local max. at $x=1$, local min. at $x=-1$
• B. local max. at $x=-1$, local min. at $x=1$
• C. local max. at $x=-1$, no local min.
• D. local min. at $x=1$ ,no local max.

Answer 1

Answer: (B)

local max. at $x=-1$, local min. at $x=1$

$f^{\prime }\left(x\right)=3x^2-3$
$=0$
Or
$x^2-1=0$
Or
$x=\pm 1$ .
$f^{\prime \prime }\left(x\right)=6x$
Hence $f^{\prime \prime }(-1)<0$ maxima.
$f^{\prime \prime }\left(1\right)>0$ ... minima.
Hence $f\left(x\right)$ attains the maximum value at $x=-1$ and a minimum value at $x=1$
$f(-1)=3-1=2$.
The maximum value is $2$