Question

The number of stationary points of F(x) = sinx in [0,$2\pi $ ] are
A)1
B)2
C)3
D)4

Answer 1

Hint: A stationary point of a function is a point where its derivative is equal to 0. At these points, the function is neither increasing nor decreasing.

Complete step by step answer:
The stationary points of a function f(x) are the values of x when $\dfrac{{df(x)}}{{dx}}$ is equal to 0
 $\dfrac{{df(x)}}{{dx}}$ = 0.
Given that ,
F(x) = sinx,
Which implies the derivative of the given function is
$\dfrac{{d(F(x))}}{{dx}}$ =$\dfrac{{d(\sin x)}}{{dx}}$ =cos(x).
Therefore the values of x when $\dfrac{{d(F(x))}}{{dx}}$=o are the stationary points.
This means we will find the values of x for which the derivative of the given function is 0. And as it is already told the function will neither be increasing or decreasing in that particular points.
=>$\dfrac{{d(F(x))}}{{dx}}$=0
=> cos(x) = 0
So now we need to find the values of x such that the above statement satisfies.
We know that the cos function will be 0 in odd intervals of $\dfrac{\pi }{2}$
=> x = $\dfrac{{(2n + 1)\pi }}{2}$ where n belongs to integers.
The values which satisfy the above statement and also in the given intervals in the question will be $\dfrac{\pi }{2},\dfrac{{3\pi }}{2}$
Therefore in the given interval [0, $2\pi $] the stationary points are $\dfrac{\pi }{2},\dfrac{{3\pi }}{2}$ .
So, the number of stationary points in the given interval [0, $2\pi $] are 2.
Hence in the given options, option B is correct.

Note:
Revise the definitions of stationary point and it’s corresponding conditions and check the interval while doing the problem. For more convenience, you can also draw the graphs and do it easily by checking the highest point and least point in the graph of the given interval.