Let $$f\left( x \right)$$ be a continuous function such that $$f\left( x \right)$$ does not vanish for all $$x\, \in \,R.$$ If $$\int_2^3 {f\left( x \right)} dx = \int_{ - 2}^3 {f\left( x \right)} dx$$ then $$f\left( x \right),\,x\, \in \,R,$$ is :
A.
an even function
B.
an odd function
C.
a periodic function
D.
none of these
Answer :
none of these
Solution :
$$\eqalign{
& \int_{ - 2}^3 {f\left( x \right)dx} - \int_2^3 {f\left( x \right)} dx = 0 \cr
& {\text{or }}\int_{ - 2}^2 {f\left( x \right)} dx = 0 \cr
& {\text{or }}\int_{ - 2}^0 {f\left( x \right)} dx + \int_0^2 {f\left( x \right)} dx = 0 \cr} $$
$${\text{or }}\int_0^2 {\left\{ {f\left( x \right) + f\left( { - x} \right)} \right\}dx = 0} ,$$ which may imply $$f\left( { - x} \right) = - f\left( x \right)$$ in $$\left[ { - 2,\,2} \right].$$
Nothing can be said for the whole of $$R.$$
Releted MCQ Question on Calculus >> Application of Integration
Releted Question 1
The area bounded by the curves $$y = f\left( x \right),$$ the $$x$$-axis and the ordinates $$x = 1$$ and $$x = b$$ is $$\left( {b - 1} \right)\sin \left( {3b + 4} \right).$$ Then $$f\left( x \right)$$ is-