If ∣ f ′′ ( x ) ∣ ≤ 1 , ∀ x ∈ R and f ( 0 ) = 0 = f ′ ( 0 ) , then which of the following can't be true ?

− 1 ≤ f ′′ ( x ) ≤ 1 . On integrating it twice with limits 0 to x , we get ∣ f ( x ) ∣ ≤ 2 x 2 ⇒ f ( ± 2 1 ) ≤ 8 1 and ∣ f ( ± 2 ) ∣ ≤ 2 Since 5 1 > 8 1 and 5 > 2 , none of the four values is possible.

Mathematics - Application of Derivative Question with Solution | TestHub

MathematicsApplication of DerivativeMonotonicity-LMVTMedium2 minQB

MathematicsMediummultiple choice

If $∣ f^{''} (x) ∣ \leq 1, \forall x \in R$ and $f (0) = 0 = f^{'} (0)$ , then which of the following can't be true ?

Options:(select one or more)

Answer:

A, B, C, D

Solution:

$- 1 \leq f^{''} (x) \leq 1$ . On integrating it twice with limits 0 to x , we get $∣ f (x) ∣ \leq \frac{x ^{2}}{2} \Rightarrow f (\pm \frac{1}{2}) \leq \frac{1}{8} and ∣ f (\pm 2) ∣ \leq 2$ Since $\frac{1}{5} > \frac{1}{8}$ and $5 > 2$ , none of the four values is possible.

Stream:JEESubject:MathematicsTopic:Application of DerivativeSubtopic:Monotonicity-LMVT

⏱ 2mℹ️ Source: QB

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