"Let f : R $\to$ R be defined as $$f(x) = \left\{ \matrix{ 2\sin \left( { - {{\pi x} \over 2}} \right),if\,x 1 \hfill \cr} \right.$$ If f(x) is continuous on R, then a + b equals :"

Question

"Let f : R $\to$ R be defined as $$f(x) = \left\{ \matrix{ 2\sin \left( { - {{\pi x} \over 2}} \right),if\,x < - 1 \hfill \cr |a{x^2} + x + b|,\,if - 1 \le x \le 1 \hfill \cr \sin (\pi x),\,if\,x > 1 \hfill \cr} \right.$$ If f(x) is continuous on R, then a + b equals :"

Accepted Answer

"$f( - {1^ - }) = 2$

$f( - {1^ + }) = |a + b - 1|$

$|a + b - 1|\, = 2$ ... (i)

$f({1^ - }) = |a + b + 1|$

$f({1^ + }) = 0$

$|a + b + 1| = 0 \Rightarrow a + b + 1 = 0$

$\Rightarrow a + b = - 1$ .... (ii)"

Let f : R $\to$ R be defined as

$$f(x) = \left\{ \matrix{ 2\sin \left( { - {{\pi x} \over 2}} \right),if\,x < - 1 \hfill \cr |a{x^2} + x + b|,\,if - 1 \le x \le 1 \hfill \cr \sin (\pi x),\,if\,x > 1 \hfill \cr} \right.$$ If f(x) is continuous on R, then a + b equals :

Solution

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Let f : R $\to$ R be defined as $$f(x) = \left\{ \matrix{ 2\sin \left( { - {{\pi x} \over 2}} \right),if\,x < - 1 \hfill \cr |a{x^2} + x + b|,\,if - 1 \le x \le 1 \hfill \cr \sin (\pi x),\,if\,x > 1 \hfill \cr} \right.$$ If f(x) is continuous on R, then a + b equals :

Solution

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More Limits, Continuity and Differentiability questions

Drill 25 more like these. Every day. Free.

Let f : R $\to$ R be defined as

$$f(x) = \left\{ \matrix{ 2\sin \left( { - {{\pi x} \over 2}} \right),if\,x < - 1 \hfill \cr |a{x^2} + x + b|,\,if - 1 \le x \le 1 \hfill \cr \sin (\pi x),\,if\,x > 1 \hfill \cr} \right.$$ If f(x) is continuous on R, then a + b equals :