Let f(x) be a non-constant twice differentiab | vedclass

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

Question

(B) Given $f(x) = f(1-x)$. Differentiating with respect to $x$,we get $f^{\prime}(x) = -f^{\prime}(1-x)$.At $x = 1/2$,$f^{\prime}(1/2) = -f^{\prime}(1

Vedclass · Accepted Answer

Vedclass · Answer

(B) Given $f(x) = f(1-x)$. Differentiating with respect to $x$,we get $f^{\prime}(x) = -f^{\prime}(1-x)$.At $x = 1/2$,$f^{\prime}(1/2) = -f^{\prime}(1-1/2) = -f^{\prime}(1/2)$,which implies $2f^{\prime}(1/2) = 0$,so $f^{\prime}(1/2) = 0$. Thus,$(B)$ is correct.Since $f(x) = f(1-x)$,the graph of $f(x)$ is symmetric about $x = 1/2$. Let $x = 1/2 + t$,then $f(1/2 + t) = f(1/2 - t)$,so $g(t) = f(1/2 + t)$ is an even function.For $(C)$,$\int_{-1/2}^{1/2} f(x+1/2) \sin x dx$. Since $f(x+1/2)$ is even and $\sin x$ is odd,their product is an odd function. The integral of an odd function over a symmetric interval $[-a, a]$ is $0$. Thus,$(C)$ is correct.For $(A)$,we have $f^{\prime}(1/4) = 0$. Since $f^{\prime}(x) = -f^{\prime}(1-x)$,$f^{\prime}(3/4) = -f^{\prime}(1/4) = 0$. Also $f^{\prime}(1/2) = 0$. By Rolle's Theorem,$f^{\prime\prime}(x)$ vanishes at least once in $(1/4, 1/2)$ and at least once in $(1/2, 3/4)$. Thus,$f^{\prime\prime}(x)$ vanishes at least twice in $[0, 1]$. Thus,$(A)$ is correct.For $(D)$,let $I = \int_{1/2}^1 f(1-t) e^{\sin \pi t} dt$. Let $1-t = u$,then $dt = -du$. When $t=1/2, u=1/2$; when $t=1, u=0$. $I = \int_{1/2}^0 f(u) e^{\sin \pi (1-u)} (-du) = \int_0^{1/2} f(u) e^{\sin \pi u} du$. Thus,$(D)$ is correct.

Similar Questions

Consider the following functions:
$I) f(x) = \begin{cases} \frac{1}{2}-x, & x < \frac{1}{2} \\ (\frac{1}{2}-x)^2, & x \geq \frac{1}{2} \end{cases}$
$II) f(x) = |3x-1|$
$III) f(x) = x|x|$
$IV) f(x) = |x|$
Then on $[0, 1]$,Lagrange's Mean Value Theorem $(LMVT)$ is applicable to which of the functions?

Let $f(1) = -2$ and $f'(x) \ge 4.2$ for $1 \le x \le 6$. The possible value of $f(6)$ lies in the interval

Let $f$ be a function that is continuous and differentiable for all real $x$. If $f(2) = -4$ and $f'(x) \geq 6$ for all $x \in [2, 4]$,then which of the following is true?

$f:[1,3] \rightarrow R$ is a function defined as $f(x)=x^3+a x^2+b x$. If $f(1)-f(3)=0$ and $f^{\prime}\left(\frac{2 \sqrt{3}+1}{\sqrt{3}}\right)=0$, then $a-b$ is equal to

Vedclass Test Series

Exam Paper Generator

Online Exam Module