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NDA 2023
Exam
NDA
Year
2023
Subject
Mathematics
Chapter
Continuity and Differentiability

NDA 2023 · Mathematics · Continuity and Differentiability

Let f(x)=\ array cc a x(x+1)+b, & x < 1 x-1 & 1 ≤ x ≤ 2 array . If...

Let f(x)=\left\{ax(x+1)+b,x<1x−11≤x≤2\begin{array}{cc}a x(x+1)+b, & x < 1 \\ x-1 & 1 \leq x \leq 2\end{array}\right. If the function f(x)f(x) is differentiable at x=1x=1, then what is the value of (a+b)(a+b) ?

Options

  1. A−13-\frac{1}{3}
  2. B-1
  3. C0
  4. D1

Correct answer

A. −13-\frac{1}{3}

Step-by-step solution

Since, \mathrm{f}(\mathrm{x})=\left\{ax(x+1)+b;x<1x−1;1≤x≤2\begin{array}{cc} \mathrm{ax}(\mathrm{x}+1)+\mathrm{b} ; & \mathrm{x} < 1 \\ \mathrm{x}-1 ; & 1 \leq \mathrm{x} \leq 2 \end{array}\right. Since, f(x)f(x) is differential at x=1x=1\mathrmf′(x)={a(2x+1);x<11;1≤x≤2⇒a(3)=1⇒a=13\begin{aligned} & \mathrmf^{\prime}(\mathrm{x})=\left\{\begin{array}{cc} \mathrm{a}(2 \mathrm{x}+1) ; & \mathrm{x} < 1 \\ 1 ; & 1 \leq \mathrm{x} \leq 2 \end{array}\right. \\ & \Rightarrow \mathrm{a}(3)=1 \Rightarrow \mathrm{a}=\frac{1}{3} \end{aligned}sincesincef(x)iscontinuousatis continuous atx=1\Rightarrow 2 \mathrm{a}+\mathrm{b}=0 \Rightarrow \mathrm{ b}=\frac{-2}{3}so,so,a+b=\frac{-1}{x}$
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