1
JEE Main 2023 (Online) 25th January Morning Shift
+4
-1

Let $$f:(0,1)\to\mathbb{R}$$ be a function defined $$f(x) = {1 \over {1 - {e^{ - x}}}}$$, and $$g(x) = \left( {f( - x) - f(x)} \right)$$. Consider two statements

(I) g is an increasing function in (0, 1)

(II) g is one-one in (0, 1)

Then,

A
Both (I) and (II) are true
B
Neither (I) nor (II) is true
C
Only (II) is true
D
Only (I) is true
2
JEE Main 2022 (Online) 29th July Morning Shift
+4
-1

Let $$f(x)=3^{\left(x^{2}-2\right)^{3}+4}, x \in \mathrm{R}$$. Then which of the following statements are true?

$$\mathrm{P}: x=0$$ is a point of local minima of $$f$$

$$\mathrm{Q}: x=\sqrt{2}$$ is a point of inflection of $$f$$

$$R: f^{\prime}$$ is increasing for $$x>\sqrt{2}$$

A
Only P and Q
B
Only P and R
C
Only Q and R
D
All P, Q and R
3
JEE Main 2022 (Online) 28th July Evening Shift
+4
-1

The function $$f(x)=x \mathrm{e}^{x(1-x)}, x \in \mathbb{R}$$, is :

A
increasing in $$\left(-\frac{1}{2}, 1\right)$$
B
decreasing in $$\left(\frac{1}{2}, 2\right)$$
C
increasing in $$\left(-1,-\frac{1}{2}\right)$$
D
decreasing in $$\left(-\frac{1}{2}, \frac{1}{2}\right)$$
4
JEE Main 2022 (Online) 28th July Morning Shift
+4
-1

If the minimum value of $$f(x)=\frac{5 x^{2}}{2}+\frac{\alpha}{x^{5}}, x>0$$, is 14 , then the value of $$\alpha$$ is equal to :

A
32
B
64
C
128
D
256
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