# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2007 | Oct-Nov | (P2-9709/02) | Q#4

Question

The equation of a curve is y = 2x tan x, where x is in radians. Find the coordinates of the stationary points of the curve for which

Solution

We are required to find the coordinates of stationary points of the curve with equation;

A stationary point on the curve is the point where gradient of the curve is equal to zero;

We can find expression for gradient of the curve and equate it with ZERO to find the coordinates of  stationary points.

Gradient (slope) of the curve is the derivative of equation of the curve. Hence gradient of curve with respect to  is:

Therefore;

Rule for differentiation of  is:

Rule for differentiation of  is:

Rule for differentiation of  is;

Since we are looking for coordinates of stationary points of the curve and gradient at stationary  point sis ZERO;

Using calculator;

 Properties of Domain Range Periodicity Odd/Even Translation/ Symmetry

We utilize the odd/even property of to find other solutions (roots) of :

Therefore;

Therefore, we have two solutions (roots) of the equation;

To find all the solutions (roots), we can utilize the periodic property of for both these values of but that will yield values beyond the desired interval of .

Hence, x-coordinates of the stationary points on the curve with equation  are;

Two values of x indicate that there are two stationary points.

Corresponding values of y coordinate can be found by substituting values of x in equation of the  curve.

 For For

Hence, coordinates of stationary points are;