Find stationary points from derivative

A question is this type if and only if it gives f'(x), asks to find f(x) from a point, and then requires finding and classifying stationary points using f'(x)=0 and f''(x).

4 questions · Moderate -0.7

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CAIE P1 2021 November Q9

12 marks Moderate -0.3

9 A curve has equation \(y = \mathrm { f } ( x )\), and it is given that \(\mathrm { f } ^ { \prime } ( x ) = 2 x ^ { 2 } - 7 - \frac { 4 } { x ^ { 2 } }\).

Given that \(\mathrm { f } ( 1 ) = - \frac { 1 } { 3 }\), find \(\mathrm { f } ( x )\).
Find the coordinates of the stationary points on the curve.
Find \(\mathrm { f } ^ { \prime \prime } ( x )\).
Hence, or otherwise, determine the nature of each of the stationary points.

CAIE P1 2022 November Q8

7 marks Moderate -0.8

8 The equation of a curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { \frac { 1 } { 2 } } - 3 x ^ { - \frac { 1 } { 2 } }\). The curve passes through the point \(( 3,5 )\).

Find the equation of the curve.
Find the \(x\)-coordinate of the stationary point.
State the set of values of \(x\) for which \(y\) increases as \(x\) increases.

CAIE P1 2014 June Q12

11 marks Moderate -0.8

12 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = x ^ { \frac { 1 } { 2 } } - x ^ { - \frac { 1 } { 2 } }\). The curve passes through the point \(\left( 4 , \frac { 2 } { 3 } \right)\).

Find the equation of the curve.
Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
Find the coordinates of the stationary point and determine its nature.

CAIE P1 2017 June Q7

8 marks Moderate -0.8

7 A curve for which \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 7 - x ^ { 2 } - 6 x\) passes through the point \(( 3 , - 10 )\).

Find the equation of the curve.
Express \(7 - x ^ { 2 } - 6 x\) in the form \(a - ( x + b ) ^ { 2 }\), where \(a\) and \(b\) are constants.
Find the set of values of \(x\) for which the gradient of the curve is positive.