Question

For every twice differentiable function $f : \mathbb R \to [-2, 2]$ with $(𝑓(0))^2 + (𝑓′(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 $x 0 \in (-4, 0)$ such that $|𝑓′(x0)| \leq 1$
• C. $\lim_{x \to \infty} f(x) = 1 $
• D. There exists $\alpha \in (-4, 4)$ such that $f (\alpha ) + f′′(\alpha) = 0$ and $𝑓′(\alpha) \neq 0$

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'(x_0) | = \left| \frac{f(0) - f (-4)}{4} \right| \leq 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) : $g(x) = f^2 (x) + (f'(x))^2$
$|f'(x_1) \leq 1 $ (by LMVT)
$|f(x_1)| \leq 2$ (given)
$\Rightarrow \, g(x_1) \leq 5 \, \, \exists x_1 \, \in (-4,0)$
Similarly $g(x_2) \leq 5 \, \, \exists x_2 \in (0,4)$
$g(0) = 85 \, \, \Rightarrow g(x)$ has maxima in $(x_1 , x_2) $ say at $\alpha$ .
$g'(\alpha) = 0 \& \, g (\alpha) \geq 85$
$2f'(\alpha) (f (\alpha) + f"(\alpha)) = 0 $
If $f'(\alpha) = 0 \, \Rightarrow g(\alpha) = f^2 (\alpha) \geq 85$ Not possible
$\Rightarrow \, \, f(\alpha) + f"(\alpha) = 0 \, \, \exists \alpha \in (x_1 , x_2) \in (-4,4) $