# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2008 | Oct-Nov | (P2-9709/02) | Q#6

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Question

Find the exact coordinates of the point on the curve   at which .

Solution

We are given that;

We are required to find;

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;

Therefore first we find;

If  and  are functions of , and if , then;

If , then;

Rule for differentiation of  is:

Rule for differentiation natural exponential function , or ;

Next we find .

Rule for differentiation of  is:

Rule for differentiation natural exponential function , or ;

If  and  are functions of , and if , then;

If , then;

As demonstrated above;

Therefore;

We are given that at the desired point;

Hence;

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

Therefore;

Hence, coordinates of the point on the curve  where  are .