1
JEE Main 2024 (Online) 27th January Evening Shift
+4
-1

Let $$g(x)=3 f\left(\frac{x}{3}\right)+f(3-x)$$ and $$f^{\prime \prime}(x)>0$$ for all $$x \in(0,3)$$. If $$g$$ is decreasing in $$(0, \alpha)$$ and increasing in $$(\alpha, 3)$$, then $$8 \alpha$$ is :

A
0
B
24
C
18
D
20
2
JEE Main 2023 (Online) 13th April Morning Shift
+4
-1

$$\max _\limits{0 \leq x \leq \pi}\left\{x-2 \sin x \cos x+\frac{1}{3} \sin 3 x\right\}=$$

A
$$\frac{5 \pi+2+3 \sqrt{3}}{6}$$
B
0
C
$$\frac{\pi+2-3 \sqrt{3}}{6}$$
D
$$\pi$$
3
JEE Main 2023 (Online) 12th April Morning Shift
+4
-1

If the local maximum value of the function $$f(x)=\left(\frac{\sqrt{3 e}}{2 \sin x}\right)^{\sin ^{2} x}, x \in\left(0, \frac{\pi}{2}\right)$$ , is $$\frac{k}{e}$$, then $$\left(\frac{k}{e}\right)^{8}+\frac{k^{8}}{e^{5}}+k^{8}$$ is equal to

A
$$e^{3}+e^{6}+e^{10}$$
B
$$e^{3}+e^{5}+e^{11}$$
C
$$e^{3}+e^{6}+e^{11}$$
D
$$e^{5}+e^{6}+e^{11}$$
4
JEE Main 2023 (Online) 11th April Morning Shift
+4
-1
Out of Syllabus

Let $$f:[2,4] \rightarrow \mathbb{R}$$ be a differentiable function such that $$\left(x \log _{e} x\right) f^{\prime}(x)+\left(\log _{e} x\right) f(x)+f(x) \geq 1, x \in[2,4]$$ with $$f(2)=\frac{1}{2}$$ and $$f(4)=\frac{1}{4}$$.

Consider the following two statements :

(A) : $$f(x) \leq 1$$, for all $$x \in[2,4]$$

(B) : $$f(x) \geq \frac{1}{8}$$, for all $$x \in[2,4]$$

Then,

A
Neither statement (A) nor statement (B) is true
B
Only statement (A) is true
C
Only statement (B) is true
D
Both the statements $$(\mathrm{A})$$ and (B) are true
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