JEE Advanced · Mathematics · 7. Trigonometry

Let $α = \sum_{k = 1}^{\infty} \sin^{2 k} (\frac{π}{6})$ . Let $g : [0, 1] \to ℝ$ be the function defined by $g (x) = 2^{a x} + 2^{a (1 - x)}$ . Then, which of the following statements is/are TRUE?

A The minimum value of $g (x)$ is $2^{\frac{7}{6}}$
B The maximum value of $g (x)$ is $1 + 2^{\frac{1}{3}}$
C The function $g (x)$ attains its maximum at more than one point
D The function $g (x)$ attains its minimum at more than one point

Verified Solution

Answer & Solution

Correct Answer

(A) The minimum value of $g (x)$ is $2^{\frac{7}{6}}$

Step-by-step Solution

Detailed explanation

Given,

α = \sum_{k = 1}^{\infty} \sin^{2 k} (\frac{π}{6})

and

g (x) = 2^{a x} + 2^{a (1 - x)}

.
Now solving,
$\alpha=\sum_{k=1}^{\infty}\left(\frac{1}{2}\right)^{2 k}=\sum_{k=1}^{\infty}\left(\frac{1}{4}\right)^k=$ $\frac{\frac{1}{4}}{1-\frac{1}{4}}=\frac{1}{3}$
Now putting the value of

α

g (x)

we get,

g (x) = 2^{\frac{x}{3}} + 2^{\frac{(1 - x)}{3}}

Now,

g^{'} (x) = \frac{\ln 2}{3} \frac{(2^{\frac{2 x}{3}} - 2^{\frac{1}{3}})}{2^{\frac{x}{3}}}

Now finding the critical point by

g^{'} (x) = 0

\Rightarrow x = \frac{1}{2}

And, derivative changes sign from negative to positive at

x = \frac{1}{2}

, hence

x = \frac{1}{2}

is point of local minimum as well as absolute minimum of

g (x)

for

x \in [0, 1]

Hence, minimum value of

g (x) = g (\frac{1}{2})

= 2^{\frac{1}{6}} + 2^{\frac{1}{6}} = 2^{\frac{7}{6}}

\Rightarrow

Option

(A)

is correct
Now maximum value of

g (x)

is either equal to

g (0)

g (1)

g (0) = 1 + 2^{\frac{1}{3}}

g (1) = 2^{\frac{1}{3}} + 1

Hence (B) and (C) are also correct.

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.

Same subject

Consider all possible permutations of the letters of the word ENDEANOEL. Match the Statements/Expressions in Column I with the Statements/Expressions in Column II.

Column I	Column II
(A)The number of permutations containing the word ENDEA, is	(p) $5 !$
(B) The number of permutations in which the letter E occurs in the first and the last positions, is	(q) $2 \times 5!$
(C) The number of permutations in which none of the letters D, L, N occurs in the last five positions, is	(r) $7 \times 5!$
(D) The number of permutations in which the letters A, E, O occur only in odd positions, is	(s) $21 \times 5!$

JEE Advanced 2008 Medium

Consider the parabola $y^2=8 x$. Let $\Delta_1$ be the area of the triangle formed by the end points of its latusrectum and the point $P\left(\frac{1}{2}, 2\right)$ on the parabola and $\Delta_2$ be the area of the triangle formed by drawing tangents at $P$ and at the end points of the latusrectum. Then, $\frac{\Delta_1}{\Delta_2}$ is

JEE Advanced 2011 Hard

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From JEE Advanced

List-I	List-II
(I)	$(P)$
(II)	$(Q)$
(III)	$(R)$
(IV)	$(S)$
	$(T)$
	$(U)$

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Let α=∑k=1∞sin2kπ6. Let g:0,1→ℝ be the function defined by gx=2ax+2a1-x. Then, which of the following statements is/are TRUE?

Step-by-step Solution

Let $α = \sum_{k = 1}^{\infty} \sin^{2 k} (\frac{π}{6})$ . Let $g : [0, 1] \to ℝ$ be the function defined by $g (x) = 2^{a x} + 2^{a (1 - x)}$ . Then, which of the following statements is/are TRUE?