Question

Which of the following statements is/are correct?
[Note: Q denotes the set of rational numbers.]

• A. If $f(x)=x^2+10\sin x$, then there exist a real number $c$ such that $f(c)=1000$.
• B. $\frac{d}{dx}\left|x^2+x\right|=|2x+1|\forall x\hat{ı}$.
• C. If $y=f(x)$ and $x=g(y)$, where $g=f^{-1}$, then $\frac{d^{2}x}{d{y}^2}=\frac{\left(\frac{d^{2}y}{d{x}^2}\right)}{{\left(\frac{dy}{dx}\right)}^3},\frac{dy}{dx}\ne 0$.
• D. Let $f:(0,5)\to R-Q$ be a continuous function such that $f(2)=\pi$, then $f(\pi )=\pi$.

Answer 1

Answer: (A), (D)

(A) Given $f(x)=x^2+10\sin x\forall x\in R$
Also, $f(0)=0$ and $f(x)$ is continuous on $R$.
$\underset{x\to \pm \infty }{\mathrm{Lim}}f(x)\to \infty \Rightarrow$ there exists some real number c such that $f(x)=1000$
(Using intermediate value theorem).
(B) Let $f(x)=\left|x^2+x\right|=|x(x+1)|$


So, $f(x)$ is non-derivable at two points viz $x=-1,0$.
(C) If $y=f(x)$ and $x=g(y)$ where $g=f^{-1}$, then
$\frac{d^{2}x}{d{y}^2}=-\frac{\frac{d^{2}y}{d{x}^2}}{{\left(\frac{dy}{dx}\right)}^3},\frac{dy}{dx}\ne 0.$
(D) As $f(x)$ is continuous on $(0,5)$ and $f(x)$ takes only irrational values such that $f(2)=\pi$, so $f(\pi )=\pi ($ As $f(x)$ must be only constant function).