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UPSC NDA Math Model Paper 2

Comprehension:(Que No. 21 - 22)

Given a function

f(x)=\begin{cases} -1, & \text{if } x \le 0 \\ ax+b, & \text{if } 0 < x < 1 \\ 1, & \text{if } x \ge 1 \end{cases}

; Where a, b are constants. The function continuous everywhere.

Question : 22 of 120

Marks: +1, -0

What is the value of b?

Solution:

Given:

f(x)=\begin{cases} -1 & \text{if } x \le 0 \\ ax+b & \text{if } 0 < x < 1 \\ 1 & \text{if } x \ge 1 \end{cases}

Given function continuous everywhere.

Let’s check the continuity at x tends to zero,

\lim\limits_{x \to 0^{+}}f(x)=\lim\limits_{x \to 0^{-}}f(x)=\lim\limits_{x \to 0}f(x)

\Rightarrow \lim\limits_{x \to 0^{-}}f(x)=\lim\limits_{x \to 0}f(x)

\Rightarrow \lim\limits_{x \to 0^{-}}-1=\lim\limits_{x \to 0}ax+b

∴ b = -1

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