Question

The curve with equation has one stationary point  in the interval . Find the exact  x-coordinate of this point.

Solution

We are required to find the x-coordinate of stationary point of the curve with equation;

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

Since it is stationary point of the curve and, hence, gradient of the curve at this point must be  ZERO.

We can find expression for gradient of the curve at this point and equate it with ZERO to find the x- coordinate of point M.

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

Therefore;

If  and  are functions of , and if , then;

If , then;

Let  and ;

We solve derivative of  and  one by one.

First we solve .

If we define , then derivative of  is;

If we have  and then derivative of  is;

Let  and , then;

Rule for differentiation of  is;

Rule for differentiation of  is:

Since ;

 Next, we solve .

Let . Then;

Rule for differentiation natural exponential function , or ;

Hence;

Now we need expression for gradient of the curve at stationary 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;

Since point is a stationary point, the gradient of the curve at this point must be equal to ZERO.

Using calculator;

Hence x-coordinate of stationary point on the curve is .