The set of all real values of lambda for whic | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) Given $f(x) = (1 - \cos^2 x)(\lambda + \sin x) = \sin^2 x(\lambda + \sin x) = \lambda \sin^2 x + \sin^3 x$.Find the derivative: $f'(x) = 2\lambda

Vedclass · Accepted Answer

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

Vedclass · Answer

(D) Given $f(x) = (1 - \cos^2 x)(\lambda + \sin x) = \sin^2 x(\lambda + \sin x) = \lambda \sin^2 x + \sin^3 x$.Find the derivative: $f'(x) = 2\lambda \sin x \cos x + 3 \sin^2 x \cos x = \sin x \cos x (2\lambda + 3 \sin x)$.For critical points,set $f'(x) = 0$. Since $\cos x 
eq 0$ for $x \in (-\frac{\pi}{2}, \frac{\pi}{2})$,we have $\sin x = 0$ or $\sin x = -\frac{2\lambda}{3}$.$x = 0$ is always a critical point. For the function to have exactly one maxima and one minima,there must be another critical point in the interval $(-\frac{\pi}{2}, \frac{\pi}{2})$.Thus,$-1 Solving $-1 Excluding $\lambda = 0$ (where the two critical points coincide at $x=0$),the set of values is $(-\frac{3}{2}, \frac{3}{2}) - \{0\}$.

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

Similar Questions

The maximum value of ${x^{1/x}}$ is

Find all points of local maxima and local minima of the function $f$ given by $f(x) = x^3 - 3x + 3$.

The equation of motion of a stone,thrown vertically upwards is $s = ut - 6.3t^2$,where the units of $s$ and $t$ are $cm$ and $sec$. If the stone reaches its maximum height in $3$ $sec$,then $u =$ ......... $cm/sec$.

If $m$ and $n$ respectively are the number of local maximum and local minimum points of the function $f(x) = \int_{0}^{x^{2}} \frac{t^{2}-5t+4}{2+e^{t}} dt$,then the ordered pair $(m, n)$ is equal to

The sum of the maximum and the minimum values of $f(x) = 3x^4 - 2x^3 - 6x^2 + 6x + 4$ in the interval $(0, 2)$ is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module