The set of all real values of for which the function f(x)= (1- ^ … - Vidyadip

MCQ

The set of all real values of $\lambda$ for which the function $f(x)=\left(1-\cos ^{2} x\right) \cdot(\lambda+\sin x)$ $x \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right),$ has exactly one maxima and exactly one minima, is

A
$\left(-\frac{1}{2}, \frac{1}{2}\right)-\{0\}$
B
$\left(-\frac{1}{2}, \frac{1}{2}\right)$
C
$\left(-\frac{3}{2}, \frac{3}{2}\right)$
✓
$\left(-\frac{3}{2}, \frac{3}{2}\right)-\{0\}$

✓

Answer

Correct option: D.

$\left(-\frac{3}{2}, \frac{3}{2}\right)-\{0\}$

d
$f(x)=\left(1-\cos ^{2} x\right)(\lambda+\sin x)$

$x \in\left(\frac{-\pi}{2}, \frac{\pi}{2}\right)$

$f(x)=\lambda \sin ^{2} x+\sin ^{3} x$

$f^{\prime}(x)=2 \lambda \sin x \cos x+3 \sin ^{2} x \cos x$

$f^{\prime}(x)=\sin x \cos x(2 \lambda+3 \sin x)$

$\sin x=0, \frac{-2 \lambda}{3}, \quad(\lambda \neq 0)$

for exactly one maxima \& minima

$\frac{-2 \lambda}{3} \in(-1,1) \Rightarrow \lambda \in\left(\frac{-3}{2}, \frac{3}{2}\right)$

$\lambda \in\left(-\frac{3}{2}, \frac{3}{2}\right)-\{0\}$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "M.C.Q (1 Marks)" from STD 12 - 6. Application of derivatives→STD 12 - 6. Application of derivatives — all question types→Mathematics STD 12 Science question papers→CBSE Board STD 12 Science question papers→CBSE Board question paper generator→

Similar questions

The point dividing the line joining the points $(1,\,\,2,\,\,3)$ and $(3,\,\, - 5,\,\,6\,)$ in the ratio $3:\, - 5$ is

Choose the correct answer from the given four options. Which of the following is the principal value branch of $\cos ^{-1} x?$

Let $f(x) = \int\limits_0^{{x^2}} {\left( {t - 1} \right)} \left( {t - 4} \right)\left( {t - 9} \right)dt$ , then

If $A$ is a square matrix of order 3 and $|2 A|=k|A|$, then find the value of $k$.

$\int_{}^{} {\frac{{{x^3}}}{{\sqrt {1 - {x^8}} }}dx = } $

The real valued function $f(x)=\frac{\operatorname{cosec}^{-1} x}{\sqrt{x-[x]}, \text { where }}$ $[ x ]$ denotes the greatest integer less than or equal to $x,$ is defined for all $x$ belonging to

Choose the correct answer from the given four option.
Integrating factor of the differential equation $(1-\text{x}^2)\frac{\text{d}\text{y}}{\text{d}\text{x}}-\text{xy}=1$is:

If $\int\text{f}(\text{x})\text{dx}=-2\cos\sqrt{\text{x}}+\text{c}$ then $f(x)$ is equal to:

The direction ratios of two lines $AB, AC$ are $1, -1, -1$ and $2, -1, 1.$ The direction ratios of the normal to the plane $\text{ABC}$ are:

Let the function, $f:[-7,0] \rightarrow R$ be continuous on $[-7,0]$ and differentiable on $(-7,0)$ If $f(-7)=-3$ and $f(x) \leq 2,$ for all $x \in(-7,0),$ then for all such functions $f, f(-1)+f(0)$ lies in the interval