# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2019 | Oct-Nov | (P2-9709/22) | Q#5

** Question**

Find the exact coordinates of the stationary point of the curve with equation

**Solution**

We are required to find the exact coordinates of the stationary point of the curve.

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

Therefore, we find the expression for gradient of the curve and equate it to ZERO.

We are given equation of the curve;

Therefore first we find from given equation of the curve.

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

Therefore;

We utilize Product Rule to differentiate .

If and are functions of , and if , then;

If , then;

Let and ;

First we differentiate .

Rule for differentiation of natural exponential function is;

Next we differentiate .

Rule for differentiation of is:

Rule for differentiation of is:

Rule for differentiation of is:

Hence;

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;

Now we need expression for gradient of the curve at point P.

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;

We know that point P is the stationary curve of the curve.

Therefore;

Single value of x indicates that there is only one stationary point.

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

Hence, coordinates of the stationary point of the curve are .

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