If $f:R \to R$ given by $f\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {2\cos x,{\rm{if}}\,\,x \le - \frac{\pi }{2}}\\ {a + \sin x + b,{\rm{if}}\,\, - \frac{\pi }{2} < x < \frac{\pi }{2}}\\ {1 + {{\cos }^2}x,{\rm{if}}\,\,x \ge \frac{\pi }{2}} \end{array}} \right.$ is a continuous Function on $R$, then $\left( {a,b} \right)$ is equal to

JEE MATHEMATICS MCQ: CONTINUITY

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Application +4 | -1

f:R \to R

given by

f\left( x \right) = \left\{ {\begin{array}{cccccccccccccccccccc} {2\cos x,\mathrm{if}\,\,x \le - \frac{\pi }{2}}\\ {a + \sin x + b,\mathrm{if}\,\, - \frac{\pi }{2} < x < \frac{\pi }{2}}\\ {1 + {{\cos }^2}x,\mathrm{if}\,\,x \ge \frac{\pi }{2}} \end{array}} \right.

is a continuous Function on

R

, then

\left( {a,b} \right)

is equal to

View answer and explanation

Correct option: A

\left( {\frac{1}{2},\frac{1}{2}} \right)

Option A (

\left( {\frac{1}{2},\frac{1}{2}} \right)

) is correct because for the function to be continuous, the left-hand limit and right-hand limit at

x = -\frac{\pi}{2}

must be equal, and only this option satisfies the condition a = b.

Practice All Chapter MCQs →

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.

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.

\{ \}

f:R \to R

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.

f

x = 0

\lambda

f

f(x) f(y)=

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

x, y \in {R}

f(2)=5

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

\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.

x=-5

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

x = 0

k