Let f(x) = Ax^3 - Bx - tan x cdot text{sgn}(x | vedclass

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

Question

(C) For $f(x)$ to be an even function,$f(x) = f(-x)$ must hold for all $x$ in the domain.Given $f(x) = Ax^3 - Bx - 	an x \cdot 	ext{sgn}(x)$.Since $

Vedclass · Accepted Answer

$2$

Vedclass · Answer

(C) For $f(x)$ to be an even function,$f(x) = f(-x)$ must hold for all $x$ in the domain.Given $f(x) = Ax^3 - Bx - 	an x \cdot 	ext{sgn}(x)$.Since $	an(-x) = -	an x$ and $	ext{sgn}(-x) = -	ext{sgn}(x)$,we have $	an(-x) \cdot 	ext{sgn}(-x) = (-	an x) \cdot (-	ext{sgn}(x)) = 	an x \cdot 	ext{sgn}(x)$.Thus,$f(-x) = A(-x)^3 - B(-x) - 	an x \cdot 	ext{sgn}(x) = -Ax^3 + Bx - 	an x \cdot 	ext{sgn}(x)$.For $f(x) = f(-x)$,we require $Ax^3 - Bx - 	an x \cdot 	ext{sgn}(x) = -Ax^3 + Bx - 	an x \cdot 	ext{sgn}(x)$.This simplifies to $2Ax^3 - 2Bx = 0$ for all $x$,which implies $A = 0$ and $B = 0$.$A = \sin^2 \alpha - \sin \alpha + \frac{1}{4} = (\sin \alpha - \frac{1}{2})^2 = 0 \Rightarrow \sin \alpha = \frac{1}{2}$.$B = 	an^2 \alpha + \frac{2}{\sqrt{3}} 	an \alpha + \frac{1}{3} = (	an \alpha + \frac{1}{\sqrt{3}})^2 = 0$ $\Rightarrow 	an \alpha = -\frac{1}{\sqrt{3}}$.We need $\alpha \in [-\frac{3\pi}{2}, 2\pi]$ such that $\sin \alpha = \frac{1}{2}$ and $	an \alpha = -\frac{1}{\sqrt{3}}$.$\sin \alpha = \frac{1}{2} \Rightarrow \alpha = \frac{\pi}{6}, \frac{5\pi}{6}, -\frac{7\pi}{6}, -\frac{11\pi}{6}$.$	an \alpha = -\frac{1}{\sqrt{3}} \Rightarrow \alpha = \frac{5\pi}{6}, -\frac{\pi}{6}, -\frac{7\pi}{6}, \frac{11\pi}{6}$.The common values are $\alpha = \frac{5\pi}{6}$ and $\alpha = -\frac{7\pi}{6}$.Thus,there are $2$ such values.

Similar Questions

Let $A=\{-1, 0, 1, 2\}$ and $B=\{-4, -2, 0, 2\}$. Let $f, g: A \rightarrow B$ be functions defined by $f(x)=x^{2}-x$ for $x \in A$ and $g(x)=2\left|x-\frac{1}{2}\right|-1$ for $x \in A$. Are $f$ and $g$ equal? Justify your answer.

Let $f(x) = x^2, x \in R$. For any $A \subseteq R$,define $g(A) = \{x \in R : f(x) \in A\}$. If $S = [0, 4]$,then which one of the following statements is not true?

If $f(x) = \frac{2^x}{2^x + \sqrt{2}}$,$x \in R$,then $\sum_{k=1}^{81} f\left(\frac{k}{82}\right)$ is equal to :

If $f: R \rightarrow R$ and $g: R \rightarrow R$ are given by $f(x)=|x|$ and $g(x)=[x]$ for each $x \in R$, then $\{x \in R: g(f(x)) \leq f(g(x))\}$ is equal to

Let $[x]$ represent the greatest integer less than or equal to $x$,${x} = x - [x]$,$\sqrt{2} = 1.414$ and $\sqrt{3} = 1.732$. If $f(x) = \{x + [\frac{x}{1+x^2}]\}$ is a real-valued function,then $f(\sqrt{2}) + f(-\sqrt{3}) = $

Vedclass Test Series

Exam Paper Generator

Online Exam Module