If f(x)=sin⁡x if x≤0x2+a2 if 0 2is continuous, on R, then a + b + ab =

f is continuous at 0⇒Ltx→0⁡f(x)=Ltx→0+⁡f(x)⇒Ltx→0-⁡sin⁡x=Ltx→0+x2+a2⇒0=a2⇒a=0f is continuous at 2⇒Ltx→2−⁡f(x)=Ltx→2+⁡f(x)⇒limx→2− (bx+2)=limx→2+ 0=2b+2=0⇒b=−1a + b + ab = 0 - 1 + 0 = -1

If f(x)=sin⁡x if x≤0x2+a2 if 0 2is continuous, on R, then a + b + ab =

f is continuous at 0⇒Ltx→0⁡f(x)=Ltx→0+⁡f(x)⇒Ltx→0-⁡sin⁡x=Ltx→0+x2+a2⇒0=a2⇒a=0f is continuous at 2⇒Ltx→2−⁡f(x)=Ltx→2+⁡f(x)⇒limx→2− (bx+2)=limx→2+ 0=2b+2=0⇒b=−1a + b + ab = 0 - 1 + 0 = -1

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If $f (x) = \{\begin{array}{ccc} \sin x & if & x \leq 0 \\ x^{2} + a^{2} & if & 0 < x < 1 \\ bx - 2 & if & 1 \leq x \leq 2 \\ 0 & if & x > 2 \end{array}$ is continuous, on R, then a + b + ab =

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Detailed Solution

f is continuous at $0 \Rightarrow \underset{x \to 0}{Lt} f (x) = \underset{x \to 0 +}{Lt} f (x) \Rightarrow \underset{x \to 0 -}{Lt} \sin x = \underset{x \to 0 +}{Lt} x^{2} + a^{2} \Rightarrow 0 = a^{2} \Rightarrow a = 0$

f is continuous at $2 \Rightarrow \underset{x \to 2 -}{Lt} f (x) = \underset{x \to 2 +}{Lt} f (x) \Rightarrow lim_{x \to 2 -} (bx + 2) = lim_{x \to 2 +} 0 = 2 b + 2 = 0 \Rightarrow b = - 1$