Let $f\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {\frac{1}{{\left| x \right|}}{\rm{for}}\,\,\left| x \right| \ge 1}\\ {a{x^2} + b\,\,{\rm{for}}\,\left| x \right| < 1} \end{array}} \right.$ If $f\left( x \right)$ is continuous and differentiable at any point, then

JEE MATHEMATICS MCQ: CONTINUITY

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Let

f\left( x \right) = \left\{ {\begin{array}{cccccccccccccccccccc} {\frac{1}{{\left| x \right|}}\mathrm{for}\,\,\left| x \right| \ge 1}\\ {a{x^2} + b\,\,\mathrm{for}\,\left| x \right| < 1} \end{array}} \right.

f\left( x \right)

is continuous and differentiable at any point, then

View answer and explanation

Correct option: B

a = - \frac{1}{2},b = \frac{3}{2}

Option B is correct because the conditions for continuity and differentiability at x = 1 lead to the equations a + b = 1 and 2a = -1, which are satisfied by a = -1/2 and b = 3/2.

More Questions from CONTINUITY

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If the function

f\left( x \right) = \left\{ {\begin{array}{cccccccccccccccccccc} {{{\left\{ {1 + \left| {\sin x} \right|} \right\}}^{\frac{a}{{\left| {\sin x} \right|}}}}, - \frac{\pi }{6} < x < 0}\\ {b,x = 0}\\ {{e^{\frac{{\tan 2x}}{{\tan 3x}}}},0 < x < \frac{\pi }{6}} \end{array}} \right.

Is continuous at

x = 0

f(x)=\left\{\begin{array}{rr}1+\cos ^{-1}\{\cot x\}, & x<\frac{\pi}{2} \\ \pi[x]+1, & x \geq \frac{\pi}{2}\end{array}\right.

Where [] denotes greatest integer and

\{ \}

denotes fractional part function, Then value of jump of discontinuity is: If

f:R \to R

is defined by

f\left( x \right) = \left\{ {\begin{array}{cccccccccccccccccccc} {\frac{{\cos 3x - \cos x}}{{{x^2}}},\mathrm{for}\,\,x \ne 0}\\ {\lambda ,\mathrm{for}\,\,x = 0} \end{array}} \right.

and if

f

is continuous at

x = 0

, then

\lambda

is equal to Let

f

be a continuous function satisfying

f(x) f(y)=

f(x)+f(y)+f(x y)-2

for all

x, y \in {R}

and

f(2)=5

then

\lim \limits_{{x \rightarrow 4}} f(x)

is If

\mathrm{f}(\mathrm{x})=\left\{\begin{array}{c}\frac{x^2+3 x-10}{x^2+2 x-15} \text {, when } x \neq-5 \\ a, \quad \text { when } x=-5\end{array}\right.

is continuous at

x=-5

, then the value of 'a' will be If

f\left( x \right) = \left\{ {\begin{array}{cccccccccccccccccccc} {x\sin \frac{1}{x},x \ne 0}\\ {k,x = 0} \end{array}} \right.

is continuous at

x = 0

, then the value of

k