Let f:R→R be a function defined as fx= 6 if x≤1 a+bx if 1<x<3 b+5x if 3≤x<5 15 if x≥5 . Then, f is

Correct answer: (D) not continuous in R for any values of a and b. Given, fx= 6 if x≤1 a+bx if 1<x<3 b+5x if 3≤x<5 15 if x≥5 Now, f1=6,f1^(-)=6,f1^(+)=a+b…..i Also, f3=b+15,f3^(-)=a+3b,f3^(+)=b+15…..ii Also, f5=15,f5^(-)=b+25,f5^(+)=15…….iii From i,ii and iii , clearly, f is not continuous for any values of a and b .

Q21 / 93 · NTA Abhyas (JEE Main) 2020 · Mathematics · Continuity and Differentiability

Let $f : R \to R$ be a function defined as $f (x) = \{\begin{matrix} 6 & i f & x \leq 1 \\ a + b x & i f & 1 < x < 3 \\ b + 5 x & i f & 3 \leq x < 5 \\ 15 & i f & x \geq 5 \end{matrix}$ . Then, $f$ is

A
continuous in $R$ if $a = - 5$ and $b = 10$
B
continuous in $R$ if $a = 5$ and $b = 5$
C
continuous in $R$ if $a = 0$ and $b = 5$
Correct answer
D
not continuous in $R$ for any values of $a$ and $b$

Solution

Given,

f (x) = \{\begin{matrix} 6 & i f & x \leq 1 \\ a + b x & i f & 1 < x < 3 \\ b + 5 x & i f & 3 \leq x < 5 \\ 15 & i f & x \geq 5 \end{matrix}

Now,

f (1) = 6, f (1^{-}) = 6, f (1^{+}) = a + b \dots . . (i)

Also,

f (3) = b + 15, f (3^{-}) = a + 3 b, f (3^{+}) = b + 15 \dots . . (i i)

Also,

f (5) = 15, f (5^{-}) = b + 25, f (5^{+}) = 15 \dots \dots . (i i i)

From

(i), (i i)

and

(i i i)

, clearly,

f

is not continuous for any values of

a

and

b

Hence, the correct answer is option: 'D'.

Let f:R→R be a function defined as fx=6ifx≤1a+bx if1<x<3b+5xif3≤x<515ifx≥5. Then, f is

Solution

More Continuity and Differentiability Questions

Let $f : R \to R$ be a function defined as $f (x) = \{\begin{matrix} 6 & i f & x \leq 1 \\ a + b x & i f & 1 < x < 3 \\ b + 5 x & i f & 3 \leq x < 5 \\ 15 & i f & x \geq 5 \end{matrix}$ . Then, $f$ is