# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2019 | May-Jun | (P2-9709/23) | Q#3

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** 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

Gradient (slope) of the curve is the derivative of equation of the curve. Hence gradient of curve

Therefore;

We utilize Quotient Rule to differentiate

If

If

Let

First we differentiate

Rule for differentiation natural logarithmic function

Next we differentiate

Rule for differentiation of

Hence;

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

Gradient (slope)

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)

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;

Taking anti-logarithm of both sides;

function, with domain the positive real numbers. 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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