Question

If $f\left(x\right)=x^5-20x^3+240x$, then $f\left(x\right)$ satisfies which of the following

• A. It is monotonically decreasing everywhere
• B. It is monotonically decreasing only in $\left(0,\infty \right)$
• C. It is monotonically increasing everywhere
• D. It is monotonically decreasing only in $\left(-\infty ,0\right)$

Answer 1

The correct option is C

It is monotonically increasing everywhere

Explanation for the correct option.

Find the nature of the function $f\left(x\right)$.

The derivative of the function $f\left(x\right)=x^5-20x^3+240x$ is given as:

$\begin{array}{rcl}f\text{'}\left(x\right)& =& 5x^4-20\left(3x^2\right)+240\\ & =& 5\left(x^4-12x^2+48\right)\\ & =& 5\left({\left(x^2\right)}^2-2x^2\times 6+6^2+12\right)\\ & =& 5\left({\left(x^2-6\right)}^2+12\right)\end{array}$

Now, square of a number is always non-negative, so ${\left(x^2-6\right)}^2\ge 0$ and thus $5\left({\left(x^2-6\right)}^2+12\right)>0$.

Thus for the function $f\left(x\right)$ it is found that $f\text{'}\left(x\right)>0$ for all $x$. So the function $f\left(x\right)$ is monotonically increasing everywhere.

Hence, the correct option is C.