Q. 12 - Calculus

Question

Prove that the function is increasing on all values of real numbers.

$f\left(x\right)=x^3$.

Step-by-Step Solution

Answer

$T\mathrm{he} \mathrm{function} \mathrm{f}\left(\mathrm{x}\right)={\mathrm{x}}^5 \mathrm{is} \mathrm{increasing} \mathrm{on} \mathrm{all} \mathrm{of} \mathrm{real} \mathrm{number}.$

Step 1. Given

The given function is $f\left(x\right)=x^3$.

Step 2. Finding the intervals

$$\begin{gathered} \mathrm{Now}, \mathrm{i}f b<a\le 0 \mathrm{then} \\ b^5 <a^5 \mathrm{Thus}, \mathrm{the} \mathrm{function} f\left(x\right)=x^5 \mathrm{is} \mathrm{increasing} \mathrm{in} \mathrm{the} \mathrm{interva}l (-\infty ,0]. \\ Now, if 0<a<b then \\ a^5<b^5 \\ \mathrm{Thus}, \mathrm{the} \mathrm{function} f\left(x\right)=x^5 \mathrm{is} \mathrm{increasing} \mathrm{in} \mathrm{the} \mathrm{interval} [0,\infty ). \\ \mathrm{Therefore}, \mathrm{the} \mathrm{function} f\left(x\right)=x^5 is \mathrm{increasing} \mathrm{in} \mathrm{the} \mathrm{interval} (-\infty ,\infty ). \end{gathered}$$

Step 3. Prove

$$\begin{gathered} \mathrm{The} \mathrm{derivative} \mathrm{of} \mathrm{the} \mathrm{function} \mathrm{is} \mathrm{given} \mathrm{by} \mathrm{f}\text{'}\left(\mathrm{x}\right)=5{\mathrm{x}}^4 \\ \mathrm{Which} \mathrm{is} \mathrm{always} \mathrm{positive} \mathrm{for} \mathrm{x}\in \mathrm{R}. \mathrm{Therefore}, \mathrm{the} \mathrm{function} \mathrm{f}\left(\mathrm{x}\right)={\mathrm{x}}^5 \mathrm{is} \mathrm{increasing} \mathrm{on} \mathrm{all} \mathrm{of} \mathrm{real} \mathrm{number}. \end{gathered}$$