For every twice differentiable function f:R→[−3,3] with (f(0))2+(f/(0))2=100 , which of the following statement(s) is (are) TRUE?

Applying LMVT on [ − 6 , 0 ] f / ( x 0 ) = f ( 0 ) − f ( − 6 ) 0 − ( − 6 ) = f ( 0 ) − f ( − 6 ) 6 for a 0 ∈ ( − 6 , 0 ) | f / ( x 0 ) | = | f ( 0 ) − f ( − 6 ) 6 | ≤ | f ( 0 ) | + | f ( − 6 ) | 6 ≤ 3 + 3 6 = 1 If f(x) is periodic then lim x → ∞ f ( x ) ≠ 1 Similarly by applying LMVT on [0, 6] | f / ( x 1 ) | ≤ 1 for x 1 ∈ ( 0 , 6 ) Consider g ( x 0 ) = ( f ( x 0 ) ) 2 + ( f / ( x 0 ) ) 2 ≤ 10 g ( x 1 ) = ( f ( x 1 ) ) 2 + ( f / ( x 1 ) ) 2 ≤ 10 Let α be the local maximum of g ( x ) ⇒ g / ( α ) = 0 , g / / ( α ) ≤ 0 ⇒ 2 f ( α ) f / ( α ) + 2 f / ( α ) f / / ( α ) = 0 ⇒ f / ( α ) ( f ( α ) + f / / ( α ) ) = 0 ⇒ f ( α ) + f / / ( α ) = 0 If f / ( α ) = 0 then g ( α ) is not local maximum of g(x) at local maximum f / ( α ) ≠ 0

There exist r , s ∈ R , where r < s such that f is one–one on the open interval (r, s)

There exist α ∈ ( − 6 , 6 ) such that f ( α ) + f / / ( α ) = 0 and f / ( α ) ≠ 0

There exists x 0 ∈ ( − 6 , 0 ) such that | f / ( x 0 ) | ≤ 1

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For every twice differentiable function $f : R \to [- 3, 3]$ with ${(f (0))}^{2} + {(f^{/} (0))}^{2} = 100$ , which of the following statement(s) is (are) TRUE?

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There exist $r, s \in R$ , where r < s such that f is one–one on the open interval (r, s)

There exist $α \in (- 6, 6)$ such that $f (α) + f^{/ /} (α) = 0$ and $f^{/} (α) \neq 0$

There exists $x_{0} \in (- 6, 0)$ such that $| f^{/} (x_{0}) | \leq 1$

$\lim_{x \to \infty} f (x) = 1$

answer is A, B, D.

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Detailed Solution

Applying LMVT on $[- 6, 0]$
$f^{/} (x_{0}) = \frac{f (0) - f (- 6)}{0 - (- 6)} = \frac{f (0) - f (- 6)}{6}$ for $a_{0} \in (- 6, 0)$
$| f^{/} (x_{0}) | = | \frac{f (0) - f (- 6)}{6} | \leq \frac{| f (0) | + | f (- 6) |}{6}$
$\leq \frac{3 + 3}{6} = 1$
If f(x) is periodic then $\lim_{x \to \infty} f (x) \neq 1$
Similarly by applying LMVT on [0, 6]
$| f^{/} (x_{1}) | \leq 1$ for $x_{1} \in (0, 6)$
Consider $g (x_{0}) = {(f (x_{0}))}^{2} + {(f^{/} (x_{0}))}^{2} \leq 10$
$g (x_{1}) = {(f (x_{1}))}^{2} + {(f^{/} (x_{1}))}^{2} \leq 10$
Let $α$ be the local maximum of $g (x) \Rightarrow g^{/} (α) = 0, g^{/ /} (α) \leq 0$
$\Rightarrow 2 f (α) f^{/} (α) + 2 f^{/} (α) f^{/ /} (α) = 0$
$\Rightarrow f^{/} (α) (f (α) + f^{/ /} (α)) = 0 \Rightarrow f (α) + f^{/ /} (α) = 0$
If $f^{/} (α) = 0$ then $g (α)$ is not local maximum of g(x) at local maximum
$f^{/} (α) \neq 0$