Question

The greatest value of ${\sin }^3\mathrm{x}+{\cos }^3\mathrm{x}$ is

• A. $1$
• B. $2$
• C. $\sqrt{2}$
• D. $\sqrt{3}$

Answer 1

The correct option is A

$1$

The explanation for the correct answer.

Find the greatest value:

$$\begin{gathered} \mathrm{y}={\sin }^3\mathrm{x}+{\cos }^3\mathrm{x} \\ \frac{\mathrm{dy}}{\mathrm{dx}}=3{\sin }^2\mathrm{xcosx}-3{\cos }^2\mathrm{xsinx} \\ \frac{\mathrm{dy}}{\mathrm{dx}}=3\mathrm{sinxcosx}(\mathrm{sinx}-\mathrm{cosx}) \\ \mathrm{Put}\frac{\mathrm{dy}}{\mathrm{dx}}=0 \\ 3\mathrm{sinxcosx}(\mathrm{sinx}-\mathrm{cosx})=0 \\ \mathrm{x}=0,\frac{\mathrm{\pi }}{2},\frac{\mathrm{\pi }}{4} \end{gathered}$$

$y$ has a maximum value at $\mathrm{x}=0\mathrm{or}\frac{\mathrm{\pi }}{2}$.

${\mathrm{y}}_{\max }=1$

Hence, option $A$ is the correct answer.