Question

The gradient of a curve at the point (x, y) is given by . The curve  has a stationary point at (a, 14), where a is a positive constant.

a)Find the value of .

b)Determine the nature of the stationary point.

c)Find the equation of the curve.

Solution

a)

We are given that gradient of the curve at the point (x, y) is;

We are also given that point at (a, 14) is a stationary point.

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

Therefore, at point (a, 14);

Gradient (slope) of the curve at the particular point is the derivative of equation of the  curve at that particular point.

Gradient (slope) of the curve at a particular point can be found by  substituting x-coordinates of that point in the expression for gradient of the curve;

Hence, gradient at point (a, 14);

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. 

Hence;

Now we have two options.

We are given that  is a positive constant.

Hence;

b)

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

We are given that gradient of the curve at the point (x, y) is;

Therefore, for second derivative;

Rule for differentiation of  is:

Rule for differentiation of  is:

Rule for differentiation f  is:

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

We substitute of the stationary point in the expression of 2^nd 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.

We are given that point (a, 14) is a stationary point on the curve and we have found  in (a) that , therefore, coordinates of given stationary point on the curve are (6,  14).

To find the nature of stationary point we substitute x-coordinate of the point in  expression of second derivative of the equation of the curve;

Since , the stationary point (6, 14) is a maximum.

 

c)

We are required to find the equation of the curve.

We can find equation of the curve from its derivative through integration;

We are given that gradient of the curve at the point (x, y) is;

Therefore;

Rule for integration of  is:

Rule for integration of  is:

Rule for integration of  is:

If a point  lies on the curve , we can find out value of . We substitute values  of  and   in the equation obtained from integration of the derivative of the  curve i.e. .

We are given that point (a, 14) lies on the curve and we have found in (a) that ,  therefore, coordinates of given point on the curve are (6, 14). 

Hence;

Hence;