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) $f^{\prime}(0)=0=f(0)$
(c) $f(0)=0$ but $f^{\prime}(0)$ not necessarily zero
(d) $|f(x)| \leq 1, x \in(0,1]$
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Answer:
Correct Answer: 15. (b)
Solution:
- Since,
$$ f^{\prime \prime}(x)=-f(x) $$
$$ \begin{array}{llll} \Rightarrow & & \frac{d}{d x}{f^{\prime}(x) } & =-f(x) \\ \Rightarrow & & g^{\prime}(x) & =-f(x) \quad\left[\because g(x)=f^{\prime}(x), \text { given }\right] \ldots(\text { (i) } \\ \text { Also, } & & F(x)=f \frac{x}{2}+g \frac{x}{2} \\ \Rightarrow & & F^{\prime}(x)=2 f \frac{x}{2} \cdot f^{\prime} \frac{x}{2} \cdot \frac{1}{2} \end{array} $$
$$ +2 \quad g \quad \frac{x}{2} \quad \cdot g^{\prime} \quad \frac{x}{2} \cdot \frac{1}{2}=0 \text { [from Eq.(i)] } $$
$$ \therefore \quad F(x) \text { is constant } \Rightarrow F(10)=F(5)=5 $$
