The set of all real values λ for which the function f(x) = (1 - ^ 2 x) . (λ + x), xε(- (π)/(2), (π)/(2)) , has exactly one maxima and exactly one minima, is :

Correct answer: D. (- (3)/(2), (3)/(2)) - \ 0\ f(x) = ^ 2 x(λ + x) f ' (x) = x x(2λ + 3 x) x = 0 (one point) x = - (2λ)/(3) ∈ (- 1, 1) - \ 0\ .....(1) λ ∈ (- (3)/(2), (3)/(2)) - \ 0\

JEE Main 20206 SeptemberEvening Shift

Exam: JEE Main
Year: 2020
Date: 6 September
Shift: Evening Shift
Subject: Mathematics
Chapter: Application of Derivatives

JEE Main 2020 · 6 September · Evening Shift · Mathematics · Application of Derivatives

The set of all real values λ for which the function f(x) = (1 - ^ 2...

The set of all real values

\lambda

for which the function

f\left(x\right) = \left(1 - \cos^{2}x\right) . \left(\lambda + \sin x\right), xε\left(- \dfrac{\pi}{2}, \dfrac{\pi}{2}\right)

, has exactly one maxima and exactly one minima, is :

Options

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

Correct answer

D. $\left(- \dfrac{3}{2}, \dfrac{3}{2}\right) - \left\{0\right\}$

Step-by-step solution

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

$f ' \left(x\right) = \sin x \cos x\left(2\lambda + 3\sin x\right)$

$\sin x = 0$

(one point)

$\sin x = - \dfrac{2\lambda}{3} \in \left(- 1, 1\right) - \left\{0\right\}$ .....(1)

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

Practice Application of Derivatives on Quantrex Academy →

More from Application of Derivatives

Consider the function f: (0, ∞) arrow (-∞, ∞) given by f(x) = x \, _e(x) - x + 1 . Then… JEE Advanced 2026 Let P be the point on the parabola y = x^2 such that the slope of the tangent to the… JEE Advanced 2026 Let f: ℝ arrow ℝ be a differentiable function such that f((x+y)/(3)) = (f(x) + f(y))/(3)… JEE Main 2026 _ 0 ≤ x ≤ π (16 ((x)/(2)) ^3((x)/(2))) is equal to: JEE Main 2026 Let f(x) be a polynomial of degree 5 , and have extrema at x = 1 and x = -1 . If _ x → 0… JEE Main 2026 The number of critical points of the function f(x) = cases |( x)/(x)|, & x ≠ 0 1, & x = 0… JEE Main 2026 Let f be a differentiable function satisfying f(x)=1-2 x+∫_ 0 ^ x e^ (x-t) f(t) dt, x ∈ R… JEE Main 2026 Consider the following three statements for the function f:(0, ∞) arrow ℝ defined by… JEE Main 2026 Let (2 α, α) be the largest interval in which the function f(t)= |t+1| t^ 2 , t 2 , is… JEE Main 2026 Let f(x)=x^ 2025 -x^ 2000 , x ∈[0,1] and the minimum value of the function f(x) in the… JEE Main 2026 Let f: ℝ → ℝ be a twice differentiable function such that f''(x) > 0 for all x ∈ ℝ and… JEE Main 2026 Let f: R arrow R be a twice differentiable function such that the quadratic equation f(x)… JEE Main 2026 Let ℝ denote the set of all real numbers. Define the function f: R arrow R by f(x)=\… JEE Advanced 2025 Let ℝ denote the set of all real numbers. For a real number x , let [x] denote the… JEE Advanced 2025 Let ℝ denote the set of all real numbers. Let f: ℝ arrow ℝ be defined by f(x)= cases (6… JEE Advanced 2025 Let the function f(x)=(x)/(3)+(3)/(x)+3, x ≠ 0 be strictly increasing in (-∞, α_1) U(α_2… JEE Main 2025 Full Application of Derivatives list Application of Derivatives JEE Maths PYQs JEE Mathematics PYQs All JEE Main PYQs