# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2015 | Oct-Nov | (P1-9709/11) | Q#5

Hits: 186

Question

A curve has equation .

i.         Find and ii.        Find the coordinates of the stationary points and state, with a reason, the nature of each                   stationary point.

Solution

i.

We are given that; Gradient (slope) of the curve is the derivative of equation of the curve. Hence gradient of curve with respect to is:    Rule for differentiation of is:  Rule for differentiation of is:      Second derivative is the derivative of the derivative. If we have derivative of the curve as , then  expression for the second derivative of the curve is;  Rule for differentiation of is:   Rule for differentiation of is: Rule for differentiation of is:   ii.

A stationary point on the curve is the point where gradient of the curve is equal to zero; From (i), we have; Coordinates of stationary point on the curve can be found from the derivative of equation of the  curve by equating it with ZERO. This results in value of x-coordinate of the stationary point on the curve.

Therefore;          Two possible values of imply that there are two stationary points on the curve one at each value of .

To find y-coordinate of the stationary point on the curve, we substitute value of x-coordinate  of the stationary point on the curve (found by equating derivative of equation of the curve  with ZERO) in the equation of the curve.

 For For       Once we have the x-coordinate of the stationary point of a curve, we can determine its  nature, whether minimum or maximum, by finding 2nd derivative of the curve.

Second derivative is the derivative of the derivative. If we have derivative of the curve as , then  expression for the second derivative of the curve is; From (i), we have; Once we have the coordinates of the stationary point of a curve,  we can determine its  nature, whether minimum or maximum, by finding 2nd derivative of the curve.

We substitute of the stationary point in the expression of 2nd derivative of the curve and  evaluate it;

If or then stationary point (or its value) is minimum.

If or then stationary point (or its value) is maximum.

 For For             Maximum Minimum