If the function f(x) , defined below, is continuous on the interval [0,8] , then f(x)=\ array cc x^2+a x+b & , 0 ≤ x < 2 3 x+2, & 2 ≤ x ≤ 4 2 a x+5 b, & 4 < x ≤ 8 array .

Correct answer: A. a=3, b=-2 Here, f(x) is continuous on the interval [0,8] . So it will also be continuous on 2 and 4. At x=2 , _ x arrow 2^ - f(x)= _ x arrow 2^ + f(x) _ x arrow 2^ - (x^2+ax+b)= _ x arrow 2^ + (3 x+2)4+2 a+b=3(2)+2 2 a+b=4 ...(i) At x=4 _ x arrow 4^ - f(x)= _ x arrow 4^ + f(x) _ x arrow 4^ - (3 x+2)= _ x arrow 4^ + 2 a x+5 b3(4)+2=2 a(4)+5 b 8 a+5 b=14 .......(ii) On solving Eqs. (i) and (ii), we get a=3 and b=-2 .

BITSAT 2024Paper 1

Exam: BITSAT
Year: 2024
Paper: Paper 1
Subject: Mathematics
Chapter: Statistics

BITSAT 2024 · Paper 1 · Mathematics · Statistics

If the function f(x) , defined below, is continuous on the interval...

If the function

f(x)

, defined below, is continuous on the interval

[0,8]

, then

f(x)=\left\{

\begin{array}{cc}x^2+a x+b & , 0 \leq x < 2 \\ 3 x+2, & 2 \leq x \leq 4 \\ 2 a x+5 b, & 4 < x \leq 8\end{array}

\right.

Options

A $a=3, b=-2$
B $a=-3, b=2$
C $a=-3, b=-2$
D $a=3, b=2$

Correct answer

A. $a=3, b=-2$

Step-by-step solution

Here,

f(x)

is continuous on the interval

[0,8]

. So it will also be continuous on 2 and 4. At

x=2

\lim _{x \rightarrow 2^{-}} f(x)=\lim _{x \rightarrow 2^{+}} f(x)\lim _{x \rightarrow 2^{-}}\left(x^2+\mathrm{ax}+\mathrm{b}\right)=\lim _{x \rightarrow 2^{+}}(3 x+2)4+2 a+b=3(2)+2\therefore 2 a+b=4

...(i) At

x=4\lim _{x \rightarrow 4^{-}} f(x)=\lim _{x \rightarrow 4^{+}} f(x)\lim _{x \rightarrow 4^{-}}(3 x+2)=\lim _{x \rightarrow 4^{+}} 2 a x+5 b3(4)+2=2 a(4)+5 b\therefore 8 a+5 b=14

.......(ii) On solving Eqs. (i) and (ii), we get

a=3

and

b=-2

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