AP EAMCET · Maths · Application of Derivatives

The difference between the greatest and least values of the function $f (x) = - x + \sin 2 x$ on $[- \frac{π}{2}, \frac{π}{2}]$ is

A $\frac{\sqrt{3} + \sqrt{2}}{?}$
B $\frac{\sqrt{3} + \sqrt{2}}{2} + \frac{π}{6}$
C $\frac{π}{2} + \frac{π}{2}$
D $\frac{\sqrt{3}}{2} - \frac{π}{6}$

Verified Solution

Answer & Solution

Correct Answer

(C) $\frac{π}{2} + \frac{π}{2}$

Step-by-step Solution

Detailed explanation

f(x)=-x+sin2x in -π2,π2 f'(x)=2cos2x-1=0 ⇒x=±π6 f(x) increasing in x∈-π6,π6 decreasing in x∈-π2,-π6∪π6,π2 f(x)maximum at x=-π2 or π6 f-π2=π2 f(x)minimum at x=π2 fπ2=-π2…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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If $f_n(x)=\frac{1}{2 n}\left[\sin ^{2 n} x+\cos ^{2 n} x\right]$, then $f_1(x)+f_2(x)-f_3$ (x) $=$

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Match the ranges of the functions given in List - I with those of the items given in List - II

List - I	List - II
(I) $3 \sin ^2 x+4 \cos ^2 x-2$	(a) $\left[\frac{1}{4}, 1\right]$
(II) $\cos ^2 x+\sin ^4 x$	(b) $\left[-\frac{1}{4}, \frac{1}{4}\right]$
(III) $\sin ^6 x+\cos ^6 x$	(c) $[1,2]$
(IV) $\cos x \cos \left(\frac{2 \pi}{3}+x\right) \cos \left(\frac{2 \pi}{3}-x\right)$	(d) $\left[\frac{3}{4}, 1\right]$
	(e) $[0,1]$

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List - I	List - II
(I) \(3 \sin ^2 x+4 \cos ^2 x-2\)	(a) \(\left[\frac{1}{4}, 1\right]\)
(II) \(\cos ^2 x+\sin ^4 x\)	(b) \(\left[-\frac{1}{4}, \frac{1}{4}\right]\)
(III) \(\sin ^6 x+\cos ^6 x\)	(c) \([1,2]\)
(IV) \(\cos x \cos \left(\frac{2 \pi}{3}+x\right) \cos \left(\frac{2 \pi}{3}-x\right)\)	(d) \(\left[\frac{3}{4}, 1\right]\)
	(e) \([0,1]\)

The difference between the greatest and least values of the function f(x)=-x+sin2x on -π2,π2 is

Step-by-step Solution

The difference between the greatest and least values of the function $f (x) = - x + \sin 2 x$ on $[- \frac{π}{2}, \frac{π}{2}]$ is