JEE Advanced 2022 · Paper 2 · Mathematics Q10

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MathematicsMultiple correct+4 / -2Question 10 of 54

Let
$\alpha = \sum_{k=1}^{\infty} \sin^{2k}\left(\frac\pi6\right).$
Let $g : [0, 1] \to \mathbb{R}$ be the function defined by
$g(x) = 2^{\alpha x} + 2^{\alpha(1-x)}.$
Then, which of the following statements is/are TRUE?

The minimum value of

g(x)

2^{\frac76}

The maximum value of

g(x)

1 + 2^{\frac13}

The function

g(x)

attains its maximum at more than one point

The function

g(x)

attains its minimum at more than one point

Answer and worked solution

Official answer: A, B, C

Find $\alpha$ . It is a geometric series with ratio $\sin^2\frac\pi6=\frac14$ :
$\alpha=\frac{1/4}{1-1/4}=\frac13,\qquad g(x)=2^{x/3}+2^{(1-x)/3}.$

Minimum. By AM-GM, $g(x)\ge2\sqrt{2^{1/3}}=2^{7/6}$ , with equality only when $x=1-x$ , i.e. $x=\frac12$ . So (A) is true and (D) is false.

Maximum. $g$ is convex and symmetric about $x=\frac12$ , so its maximum on $[0,1]$ is at both ends:
$g(0)=g(1)=1+2^{1/3}.$
(B) and (C) are true. (A), (B), (C).

Topics:Maxima and minima Geometric progression AM-GM-HM inequalities

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JEE Advanced 2022 · Paper 2, Mathematics question 10