Question

Which of the following functions has neither local maxima nor local minima?

• A. $f\left(x\right)=x^2+x$
• B. $f\left(x\right)=\log x$
• C. $f\left(x\right)=x^3-3x+3$
• D. $f\left(x\right)=3+\left|x\right|$

Answer 1

The correct option is B $f\left(x\right)=\log x$

Option A:

$f\left(x\right)=x^2+x$

$\Rightarrow f^{\prime }\left(x\right)=2x+1$

For $x=-\frac{1}{2},f^{\prime }\left(x\right)=0$

Hence, $x^2+x$ has either a local maxima or minima.

Option B:

$f\left(x\right)=\log x$

$\Rightarrow f^{\prime }\left(x\right)=\frac{1}{x}$

$f^{\prime }\left(x\right)\ne 0\forall x\in R$

Hence, $\log x$ has neither a local maxima nor minima.

Option C:

$f\left(x\right)=x^3-3x+3$

$\Rightarrow f^{\prime }\left(x\right)=3x^2-3$

For $x=\{-1,1\},f^{\prime }\left(x\right)=0$

Hence, $x^3-3x+3$ has either a local maxima or minima.

Option D:

$f\left(x\right)=\{\begin{array}{c}\begin{array}{cc}3+x& x\ge 0\\ 3-x& x<0\end{array}\end{array}$

From the function itself, it can be seen that $f\left(x\right)$ has a sharp point at $x=0$ which is a point of minima.

Hence, $3+|x|$ has a local minima.

Option $B$ is the correct answer.