If fx=x^2+α for x≥0 =2√(x^2+1)+β for x<0 is continuous at x=0 and f1/2=2 then α^2+β^2 is

Correct answer: (C) 25/8. lim x→ 0^(+) x^2 +α= lim x→ 0^(-) 2√(x^2 +1) +β α=2+β.........( 1 ) f( x )=x^2 +α; x≥0 f( 1/2 )=1/4 +α 2=1/4 +α 7/4 =α 7/4 −2=β ⇒ β = - 1/4 Hence α^2 +β^2 =49/16 +1/16 =50/16 =25/8

Q91 / 150 · MHT CET 2018 · Mathematics

If $f (x) = x^{2} + α$ for $x \geq 0$ $= 2 \sqrt{x^{2} + 1} + β$ for $x < 0$ is continuous at $x = 0$ and $f (\frac{1}{2}) = 2$ then $α^{2} + β^{2}$ is

A
$3$
B
$\frac{8}{25}$
Correct answer
C
$\frac{25}{8}$
D
$\frac{1}{3}$

Solution

\lim_{x \to 0^{+}} x^{2} + α = \lim_{x \to 0^{-}} 2 \sqrt{x^{2} + 1} + β

α = 2 + β ......... (1)

f (x) = x^{2} + α; x \geq 0

f (\frac{1}{2}) = \frac{1}{4} + α

2 = \frac{1}{4} + α

\frac{7}{4} = α

\frac{7}{4} - 2 = β

\Rightarrow β = - \frac{1}{4}

Hence

α^{2} + β^{2} = \frac{49}{16} + \frac{1}{16} = \frac{50}{16} = \frac{25}{8}

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

If fx=x2+α for x≥0 =2x2+1+β for x<0 is continuous at x=0 and f12=2 then α2+β2 is

Solution

Relevant Questions

If $f (x) = x^{2} + α$ for $x \geq 0$ $= 2 \sqrt{x^{2} + 1} + β$ for $x < 0$ is continuous at $x = 0$ and $f (\frac{1}{2}) = 2$ then $α^{2} + β^{2}$ is