Question

For every twice differentiable function $f:\mathbb{R}\to [-2,2]$ with $(f(0){)}^2+(f\prime (0){)}^2=85$ which of the following statement(s) is (are) TRUE?

• A. There exist $r,s\in \mathbb{R}$, where $r<s$, such that $f$ is one-one on the open interval $(r,s)$
• B. There exists $x0\in (-4,0)$ such that $|f\prime (x0)|\le 1$
• C. ${\lim }_{x\to \infty }f(x)=1$
• D. There exists $\alpha \in (-4,4)$ such that $f(\alpha )+f\prime \prime (\alpha )=0$ and $f\prime (\alpha )\ne 0$

Answer 1

Answer: (A), (B), (D)

$f(x)$ can't be constant throughout the domain. Hence we can find $x\in (r,s)$ such that $f(x)$ is one-one option (A) is true.
Option (B) : $f^{\prime }(x_0)|=\left|\frac{f(0)-f(-4)}{4}\right|\le 1$ (LMVT)
Option (C) : $f(x)=\sin (\sqrt{85}x)$ satisfies given condition.
But ${\lim }_{x\to \infty }\sin (\sqrt{85})$ D.N.E.
$\Rightarrow$ Incorrect
Option (D) : $g(x)=f^2(x)+(f^{\prime }(x){)}^2$
$|f^{\prime }(x_1)\le 1$ (by LMVT)
$|f(x_1)|\le 2$ (given)
$\Rightarrow g(x_1)\le 5\exists x_1\in (-4,0)$
Similarly $g(x_2)\le 5\exists x_2\in (0,4)$
$g(0)=85\Rightarrow g(x)$ has maxima in $(x_1,x_2)$ say at $\alpha$ .
$g^{\prime }(\alpha )=0\&g(\alpha )\ge 85$
$2f^{\prime }(\alpha )(f(\alpha )+f"(\alpha ))=0$
If $f^{\prime }(\alpha )=0\Rightarrow g(\alpha )=f^2(\alpha )\ge 85$ Not possible
$\Rightarrow f(\alpha )+f"(\alpha )=0\exists \alpha \in (x_1,x_2)\in (-4,4)$