Find curve from second derivative

A question is this type if and only if it gives d²y/dx² and information about a stationary point or gradient, requiring integration twice to find the curve equation.

4 questions · Moderate -0.4

1.08b Integrate x^n: where n != -1 and sums

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CAIE P1 2019 June Q10

9 marks Moderate -0.8

10 A curve for which $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 2 x - 5$ has a stationary point at $( 3,6 )$.

Find the equation of the curve.
Find the $x$-coordinate of the other stationary point on the curve.
Determine the nature of each of the stationary points. \includegraphics[max width=\textwidth, alt={}, center]{ebf16cae-1e80-44d2-9c51-630f5dc3c11f-20_700_616_262_762} The diagram shows part of the curve $y = \frac { 3 } { \sqrt { ( 1 + 4 x ) } }$ and a point $P ( 2,1 )$ lying on the curve. The normal to the curve at $P$ intersects the $x$-axis at $Q$.
Show that the $x$-coordinate of $Q$ is $\frac { 16 } { 9 }$.
Find, showing all necessary working, the area of the shaded region.
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

Edexcel P1 2020 January Q11

11 marks Standard +0.3

11. A curve has equation $y = \mathrm { f } ( x )$, where $$f ^ { \prime \prime } ( x ) = \frac { 6 } { \sqrt { x ^ { 3 } } } + x \quad x > 0$$ The point $P ( 4 , - 50 )$ lies on the curve.
Given that $\mathrm { f } ^ { \prime } ( x ) = - 4$ at $P$,

find the equation of the normal at $P$, writing your answer in the form $y = m x + c$, where $m$ and $c$ are constants,
(3)
find $\mathrm { f } ( x )$.
(8)
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Edexcel P1 2019 October Q11

10 marks Moderate -0.8

A curve has equation $y = \mathrm { f } ( x )$.

The point $P \left( 4 , \frac { 32 } { 3 } \right)$ lies on the curve.
Given that

$\mathrm { f } ^ { \prime \prime } ( x ) = \frac { 4 } { \sqrt { x } } - 3$
$\quad \mathrm { f } ^ { \prime } ( x ) = 5$ at $P$ find
1. the equation of the tangent to the curve at $P$, writing your answer in the form $y = m x + c$, where $m$ and $c$ are constants to be found,
2. $\mathrm { f } ( x )$.

OCR C2 2015 June Q5

7 marks Moderate -0.3

5 A curve has an equation which satisfies $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 3 x ^ { - \frac { 1 } { 2 } }$ for all positive values of $x$. The point $P ( 4,1 )$ lies on the curve, and the gradient of the curve at $P$ is 5 . Find the equation of the curve.