Differentiation (Cambridge (CIE) IGCSE Maths: Extended): Exam Questions

A curve has the equation  $y=x^3+8x^2+5x.$

Work out the coordinates of the two turning points.

(.................... , ....................) and (.................... , ....................)

Determine whether each of the turning points is a maximum or a minimum. Give reasons for your answers.

A curve has equation $y=4x^3-3x+3$.

i) Find the coordinates of the two stationary points.  

( .................... , .................... ) and ( .................... , .................... ) [5]

ii) Determine whether each of the stationary points is a maximum or a minimum. Give reasons for your answers.  

[3]

A curve has equation $y=x^3–6x^2+16$.

Find the coordinates of the two turning points.  

(............ , ............) and (............ , ............)

Determine whether each of the turning points is a maximum or a minimum.

Give reasons for your answers.

The diagram above shows a sketch of a curve with equation $y=2x^3-2x^2-3x+5$.

The point $P$ is the turning point of the curve. The $x$-coordinate of $P$ is positive.

Find the $x$-coordinate of the point $P$.

Using the second derivative, show that the point $P$ is a local minimum.

$y=5x^m+p{x}^{12}$ and $\frac{\mathrm{d}y}{\mathrm{d}x}=30x^{m-1}+6x^{11}$

Find the value of $m$ and the value of $p$.

$m=$ ....................................................

$p=$ ....................................................

The curve $\mathbf{C}$ has equation $y=a{x}^3+b{x}^2–12x+6$ where $a$ and $b$ are constants.

The point $A$ with coordinates (2, –6) lies on $\mathbf{C}$.

The gradient of the curve at $A$ is 16.

Find the $y$ coordinate of the point on the curve whose $x$ coordinate is 3.

Show clear algebraic working.

Part of the graph with equation $y=2x^4-16x^2+3$ is shown below.

The graph has three stationary points, indicated on the graph by points P , Q and R. Find the area of the triangle PQR.

The curve $\mathbf{C}$ has equation $y=5x^3–x^2–6x+4$.

Find  $\frac{\mathrm{d}y}{\mathrm{d}x}$.

   $\frac{\mathrm{d}y}{\mathrm{d}x}$ = ..............................................

There are two points on the curve $\mathbf{C}$ at which the gradient of the curve is $2$.

Find the $x$ coordinate of each of these two points. Show clear algebraic working.