If f(x) = sin^2 x + x sin 2x log x,then f(x) | vedclass

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

Question

(B) Given $f(x) = \sin^2 x + x \sin 2x \log x$. We can rewrite $f(x)$ as $f(x) = x \left( \frac{\sin^2 x}{x} + \sin 2x \log x \right)$. This does not

Vedclass · Accepted Answer

at least two roots in $(0, 2\pi]$

Vedclass · Answer

(B) Given $f(x) = \sin^2 x + x \sin 2x \log x$. We can rewrite $f(x)$ as $f(x) = x \left( \frac{\sin^2 x}{x} + \sin 2x \log x \right)$. This does not simplify directly,so let us consider the function $g(x) = \sin^2 x \log x$. Then $g'(x) = 2 \sin x \cos x \log x + \sin^2 x \cdot \frac{1}{x} = \sin 2x \log x + \frac{\sin^2 x}{x}$. Thus,$f(x) = x g'(x)$. We observe that $g(1) = \sin^2(1) \log(1) = 0$. Also,$g(\pi) = \sin^2(\pi) \log(\pi) = 0$ and $g(2\pi) = \sin^2(2\pi) \log(2\pi) = 0$. By Rolle's Theorem,$g'(x) = 0$ has at least one root in $(1, \pi)$ and at least one root in $(\pi, 2\pi)$. Since $f(x) = x g'(x)$,$f(x) = 0$ implies $g'(x) = 0$ for $x \in (0, 2\pi]$. Therefore,$f(x) = 0$ has at least two roots in $(1, 2\pi) \subset (0, 2\pi]$.

If $f(x) = \sin^2 x + x \sin 2x \log x$,then $f(x) = 0$ has

Similar Questions

If $2a + 3b + 6c = 0$ and $a, b, c \in \mathbb{R}$,then the equation $ax^2 + bx + c = 0$ has at least one root between $0$ and $1$.

For which real value of $K$ does the equation $2x^3 + 3x + K = 0$ have two real roots in the interval $[0, 1]$?

If $a + b + c = 0$,then how many roots does the equation $3ax^2 + 2bx + c = 0$ have in the interval $(0, 1)$?

Let $R$ be the set of all real numbers and $f:[-1,1] \rightarrow R$ be defined as $f(x) = \begin{cases} x \sin \frac{1}{x}, & x \neq 0 \\ 0, & x=0 \end{cases}$. Then:

Consider the quadratic equation $ax^2+bx+c=0$,where $2a+3b+6c=0$ and let $g(x)=\frac{ax^3}{3}+\frac{bx^2}{2}+cx$.
Statement-$I$ : The given quadratic equation $ax^2+bx+c=0$ has at least one root in $(0,1)$.
Statement-$II$ : Rolle's theorem is applicable to $g(x)$ on $[0,1]$.
Then

Vedclass Test Series

Exam Paper Generator

Online Exam Module