Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2012 | Oct-Nov | (P2-9709/22) | Q#2
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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 .
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