Let f: R → R be a function defined as f(x) = \ array ll 5, ifx ≤ 1 a + bx, if1 Then f is:

Correct answer: C. not continuous for any values of a and b f(x) = \ array ll 5: x ≤ 1 a + bx: 1 For f ( x ) to be continuous at x = 5 _ x → 5 ^ - f(x) = _ x → 5 ^ + f(x) = f(5) ⇒ _ x → 5 ^ - (b + 5x) = 30 ⇒ b + 25 = 30 ⇒ b = 5 For f ( x ) to be continuous at x = 1 _ x → 1 ^ + f(x) = _ x → 1 ^ - f(x) = f(1) ⇒ _ x → 1 ^ + (a + 5x) = 5 ⇒ a = 0 Now at x = 3, LHL = _ x → 3 ^ - (0 + 5x) = 15 RHL = _ x → 3 ^ + (5 + 5x) = 20 Hence function f ( x ) is discontinuous at x = 3 for all a, b ∈ R .

JEE Main 20199 JanuaryMorning Shift

Exam: JEE Main
Year: 2019
Date: 9 January
Shift: Morning Shift
Subject: Mathematics
Chapter: Continuity and Differentiability

JEE Main 2019 · 9 January · Morning Shift · Mathematics · Continuity and Differentiability

Let f: R → R be a function defined as f(x) = \ array ll 5, ifx ≤ 1...

Let

f: R \to R be a function defined as<br /><br />

f\left(x\right) = \left\{

\begin{array}{ll}5, ifx \le 1 \\ a + bx, if1 < x < 3 \\ b + 5x, if3 \le x < 5 \\ 30, ifx \ge 5\end{array}

\right)

Then

f

is:

Options

Acontinuous if $a = - 5$ and $b = 10$
Bcontinuous if $a = 0$ and $b = 5$
Cnot continuous for any values of $a$ and $b$
Dcontinuous if $a = 5$ and $b = 5$

Correct answer

C. not continuous for any values of $a$ and $b$

Step-by-step solution

f\left(x\right) = \left\{

\begin{array}{ll}5: x \le 1 \\ a + bx: 1 < x < 3 \\ b + 5x: 3 \le x < 5 \\ 30: x \ge 5\end{array}

\right)

For

f ( x )

to be continuous at

x = 5

\lim_{x \to {5}^{-}}f\left(x\right) = \lim_{x \to {5}^{+}}f\left(x\right) = f\left(5\right)

⇒ \lim_{x \to {5}^{-}}\left(b + 5x\right) = 30

⇒ b + 25 = 30 ⇒ b = 5

For

f ( x )

to be continuous at

x = 1

\lim_{x \to {1}^{+}}f\left(x\right) = \lim_{x \to {1}^{-}}f\left(x\right) = f\left(1\right)

⇒ \lim_{x \to {1}^{+}}\left(a + 5x\right) = 5

⇒ a = 0

Now at

x = 3,

LHL = \lim_{x \to {3}^{-}}\left(0 + 5x\right) = 15

RHL = \lim_{x \to {3}^{+}}\left(5 + 5x\right) = 20

Hence function

f ( x )

is discontinuous at

x = 3

for all

a, b \in R

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