Question

Assertion :Statement 1: The value of $x$ for which $(\sin x+\cos x{)}^{1+\sin 2x}=2$ when $0\le x\le \pi$ is $\frac{\pi }{4}$ only Reason: Statement 2: The maximum value of $\sin x+\cos x$ occurs when $x=\frac{\pi }{4}$

• A. if both the statements are TRUE and STATEMENT 2 is the correct explanation of STATEMENT 1
• B. if both the statements are TRUE and STATEMENT 2 is NOT the correct explanation of STATEMENT 1
• C. if STATEMENT 1 is TRUE and STATEMENT 2 is FALSE
• D. if STATEMENT 1 is FALSE and STATEMENT 2 is TRUE

Answer 1

The correct option is A if both the statements are TRUE and STATEMENT 2 is the correct explanation of STATEMENT 1

Let $D=\sin x+\cos x$

$\frac{dD}{dx}=\cos x-\sin x=0$

$\cos x=\sin x$

$\tan x=1=\tan \frac{\pi }{4}$

$x=n\pi +\frac{\pi }{4}$

in $0\le x\le 4$

$x=\frac{\pi }{4}$

$[\frac{d^{2}x}{d{x}^2}{]}_{x=\frac{\pi }{4}}=-\sin x-\cos x<0$

$\therefore \sin x+\cos x$ is maximum at $\frac{\pi }{4}$

$\therefore$ Statement 2 is correct.

$M=(\sin x+\cos x{)}^{1+\sin 2x}$

$=(\sin x+\cos x)(\sin x+\cos x{)}^2$

If $x=\frac{\pi }{4}$

$M=(\sqrt{2}{)}^2=2$

$\therefore$ Statement 1 is correct.

Statement 2 is the correct reason for Statement 1 because in that case only the value of $(\sin x+\cos x{)}^{1+\sin 2x}$ is maximum.