SATHEE: Limit Continuity and Differentiability 7 Question 15

Limit Continuity and Differentiability 7 Question 15

15. If $f$ is a differentiable function satisfying

$f \frac{1}{n} = 0, \forall n \geq 1, n \in I$ , then

$(2005, 2 M)$

(a) $f (x) = 0, x \in (0, 1]$

(b) $f^{'} (0) = 0 = f (0)$

(d) $| f (x) | \leq 1, x \in (0, 1]$

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Answer:

Correct Answer: 15. (b)

Solution:

Since,

$f^{''} (x) = - f (x)$

$\begin{array}{llll} \Rightarrow & \frac{d}{d x} f^{'} (x) & = - f (x) \\ \Rightarrow & g^{'} (x) & = - f (x) [∵ g (x) = f^{'} (x), given] \dots ((i) \\ Also, & F (x) = f \frac{x}{2} + g \frac{x}{2} \\ \Rightarrow & F^{'} (x) = 2 f \frac{x}{2} \cdot f^{'} \frac{x}{2} \cdot \frac{1}{2} \end{array}$

$+ 2 g \frac{x}{2} \cdot g^{'} \frac{x}{2} \cdot \frac{1}{2} = 0 [from Eq.(i)]$

$∴ F (x) is constant \Rightarrow F (10) = F (5) = 5$

Limit Continuity and Differentiability 7 Question 15

15. If $f$ is a differentiable function satisfying

Answer:

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Limit Continuity and Differentiability 7 Question 15

15. If f is a differentiable function satisfying

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

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15. If $f$ is a differentiable function satisfying