Question

Which of the following function has extreme point(s)

• A. All of the above.
• B. $x-\left[x\right],\text{where}[.]\text{denotes the greatest integer function.}$
• C. ${\log }_{10}x$
• D. $2^x$

Answer 1

The correct option is B $x-\left[x\right],\text{where}[.]\text{denotes the greatest integer function.}$

For $y=2^x$
$\Rightarrow \frac{dy}{dx}=2^x\cdot \ln 2=0$ has no solution.

For $y={\log }_{10}x$
$\Rightarrow \frac{dy}{dx}=\frac{1}{x}\cdot {\log }_{e}10=0$ has no solution

For $y=x-\left[x\right]=\left\{x\right\}$;

From graph, we can see that every integral point is the point of local minima.