Limit Continuity and Differentiability 7 Question 1
1. Let $f: R \rightarrow R$ be differentiable at $c \in R$ and $f(c)=0$. If $g(x)=|f(x)|$, then at $x=c, g$ is
(2019 Main, 10 April I)
(a) not differentiable
(b) differentiable if $f^{\prime}(c) \neq 0$
(c) not differentiable if $f^{\prime}(c)=0$
(d) differentiable if $f^{\prime}(c)=0$
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Answer:
Correct Answer: 1. (b)
Solution:
- We know
$$ (1+x)^{n}={ }^{n} C _0+{ }^{n} C _1 x+{ }^{n} C _2 x^{2}+\ldots+{ }^{n} C _n x^{n} $$
On differentiating both sides w.r.t. $x$, we get $n(1+x)^{n-1}={ }^{n} C _1+2{ }^{n} C _2 x+\ldots+n{ }^{n} C _n x^{n-1}$
On multiplying both sides by $x$, we get
$$ n x(1+x)^{n-1}={ }^{n} C _1 x+2{ }^{n} C _2 x^{2}+\ldots+n^{n} C _n x^{n} $$
Again on differentiating both sides w.r.t. $x$,
we get
$n\left[(1+x)^{n-1}+(n-1) x(1+x)^{n-2}\right]$
$$ ={ }^{n} C _1+2^{2}{ }^{n} C _2 x+\ldots+n^{2}{ }^{n} C _n x^{n-1} $$
Now putting $x=1$ in both sides, we get
$$ \begin{gathered} { }^{n} C _1+\left(2^{2}\right){ }^{n} C _2+\left(3^{2}\right){ }^{n} C _3+\ldots+\left(n^{2}\right){ }^{n} C _n \\ =n\left(2^{n-1}+(n-1) 2^{n-2}\right) \end{gathered} $$
For $n=20$, we get
$$ \begin{aligned} & { }^{20} C _1+\left(2^{2}\right){ }^{20} C _2+\left(3^{2}\right){ }^{20} C _3+\ldots+(20){ }^{2}{ }^{20} C _{20} \\ & \quad=20\left(2^{19}+(19) 2^{18}\right) \\ & \quad=20(2+19) 2^{18}=420\left(2^{18}\right) \\ & \quad=A\left(2^{B}\right) \text { (given) } \end{aligned} $$
On comparing, we get
$$ (A, B)=(420,18) $$
