Question 326 of 401EasyMCQ
If the function \[f(x) = \left\{ {(a∣π−x∣+1,x≤5b∣x−π,+3,x>5) } \right.\]
is
continuous at x = 5, then the value of a -- b is :-
Solutions · 1
Official solution
As f(x) is continuous at x = 5 then
\[\mathop {\lim }\limits_{x \to {5^ - }} f\left( x \right) = f\left( 5 \right) = \mathop {\lim }\limits_{x \to {5^ + }} f\left( x \right)\]
\[ \Rightarrow \] \[\mathop {\lim }\limits_{h \to 0} f\left( {5 - h} \right) = f\left( 5 \right) = \mathop {\lim }\limits_{h \to 0} f\left( {5 + h} \right)\]
\[ \Rightarrow \] \[\mathop {\lim }\limits_{h \to 0} \left( {a\left| {\pi - \left( {5 - h} \right)} \right| + 1} \right) = a\left| {\pi - 5} \right| + 1\]
= \[\mathop {\lim }\limits_{h \to 0} \left( {b\left| {5 + h - \pi } \right| + 3} \right)\]
\[ \Rightarrow \] \[{a\left| {\pi - 5} \right| + 1}\] = \[a\left| {\pi - 5} \right| + 1\]
= \[{b\left| {5 - \pi } \right| + 3}\]
Note : As \[\pi \] = 3.14, then \[\pi \] - 5 = 3.14 - 5 = -1.84 < 0
\[ \therefore \] \[{\left| {\pi - 5} \right|}\] = - (\[\pi \] - 5) and \[{\left| {5 - \pi } \right|}\] = (5 - \[\pi \])
\[ \Rightarrow \] \[ - a\left( {\pi - 5} \right) + 1\] = \[ - a\left( {\pi - 5} \right) + 1\] = \[{b\left( {5 - \pi } \right) + 3}\]
\[ \Rightarrow \] \[ - a\left( {\pi - 5} \right) + 1\] = \[{b\left( {5 - \pi } \right) + 3}\]
\[ \Rightarrow \] \[\left( {a - b} \right)\left( {5 - \pi } \right) = 2\]
\[ \Rightarrow \] \[\left( {a - b} \right) = {2 \over {\left( {5 - \pi } \right)}}\]
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