GATE DS&AI 2024 | Question: 27

Question

​​​​​Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function. Note: $\mathbb{R}$ denotes the set of real numbers.

\[
f(x)=\left\{\begin{array}{cl}
-x, & \text { if } x<-2 \\
a x^{2}+b x+c, & \text { if } x \in[-2,2] \\
x, & \text { if } x>2
\end{array}\right.
\]

Which ONE of the following choices gives the values of $a, b, c$ that make the function $f$ continuous and differentiable?

• A. $a=\frac{1}{4}, b=0, c=1$
• B. $a=\frac{1}{2}, b=0, c=0$
• C. $a=0, b=0, c=0$
• D. $a=1, b=1, c=-4$

Comments

As $f(x)$ is continuous over its domain.

So, $(1)$  At $x=-2,$ $f(-2^-)=f(-2)$

$4a-2b+c=2$ $\;\;\; (1)$

$(2)$  At $x=2,$ $f(2^+)=f(2)$

$4a+2b+c=2$ $\;\;\; (2)$

From $(1)$ and $(2)$

$b=0$

Also, As $f(x)$ is differentiable over its domain.

So, at $x=2,$   $f'(2^+)=f'(2)$

$4a+b=1$

Since $b=0$ and so, $a=\frac{1}{4}$ and so from $(1)$ $c=1$

Therefore,

$a=1/4;b=0;c=1$

Answer 1

option A