Application of Derivatives — Maths STD 12 Science — Question

1. $\frac{\text{x}}{1+\text{x}^2}$
   
2. $\text{x}\sqrt{1+\text{x}^2}$
   
3. $\frac{\text{x}}{\sqrt{1+\text{x}^2}}$
   
4. $\frac{\text{x}}{\sqrt{1+\text{x}^2}}$
   

$ 2 x+7 y+\lambda z=3 $

$ 3 x+2 y+5 z=4 $

$ x+\mu y+32 z=-1$

has infinitely many solutions, then $(\lambda-\mu)$ is equal to $\qquad$

1. a = 2
2. a = 1
3. a = 0
4. $\text{a}=\frac{1}{2}$.

$\alpha=\sum_{ k =1}^{\infty} \sin ^{2 k}\left(\frac{\pi}{6}\right)$

Let $g:[0,1] \rightarrow R$ be the function defined by

$g( x )=2^{\alpha x }+2^{\alpha(1- x )}$

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

1. $\text{g(x)}=\Big(\frac{\text{b}-\text{x}^\frac{1}{3}}{\text{a}}\Big)^\frac{1}{2}$
   
2. $\text{g(x)}=\frac{1}{(\text{ax}^2+\text{b})^3}$
   
3. $\text{g(x)}=(\text{ax}^2+\text{b})^\frac{1}{3}$
   
4. $\text{g(x)}=\Big(\frac{\text{x}^\frac{1}{3}+\text{b}}{\text{a}}\Big)^\frac{1}{2}$
   

1. $\log(\log\text{x})$   
2. $\text{e}^{\text{x}}$
3. $\log\text{x}$
4. $\text{x}$