On the following page, we will look at graphs and derivatives. We will get some practice in sketching gradient functions and we will carefully consider stationary points (maximum, minimum and points of inflexion) as well as non-stationary points of inflexion.
On this page, you should learn about
- graphs of \(f,f'\ \mathrm{and} \ f''\)
- local maximum and minimum points
- stationary points of inflexion
- non-stationary points of inflexion
The following videos will help you understand all the concepts from this page
In the following video we shall see how to sketch graphs of a gradient function
You can get some practice of sketching gradient functions by trying the activity below. This works best if you have a device with a tactile screen so that you can sketch the graphs on screen. If this is not the case, you can print a copy from here and draw with a pencil.
- Choose the pen tool and sketch the gradient function
- Check your answer by clicking in the check box
- Turn off the answer and delete your sketch before moving on to the next question
Print from here
Here is a quiz that practises the skills from this page
START QUIZ!
The following diagram shows the graph fo the function f
How many stationary points does f have?
Stationary points when gradient of the function = 0
The following diagram show a graph of f' , the derivative of f
How many stationary points does f have?
Stationary points when \(f' (x) = 0\)
The following diagram show a graph of f' , the derivative of f
How many stationary points does f have?
There is no value of x for which \(f' (x) = 0\)
The diagram below is a graph of f
Fill in the missing parts in the following sentences.
| a. At the point A, the function is | a point of inflexion |
| b. At the point B, there is | even increasing |
| c. At point C, there is | a local minimum equal to zero |
| d. Between A and B the function is | negative concave up |
| e. Between C and D the function is | a local maximum |
The following diagram shows the graph of the function f
Fill in the blanks below.
- If more than one answer is required put your answers in alphabetical order
- If an interval is required, put the largest possible interval
a. The gradient is zero at and
b. There is a point of inflexion at
c. The function is negative between and
d. The function is increasing between and
e. The function is concave up (positive concavity) between and
f. The function is concave down (negative concavity) between and
g. The solution to f(x) = 0 is at
h. The solution to f(0) is at
i. The solution to f'(x) = 0 is at and
j. f'(x) > 0 between and
a. Gradient = 0 when graph of f is horizontal
b. D is a non-stationary point of inflexion
c. Graph is below the x axis
d. Gradient of f is positive
e. Gradient of f is increasing. Think about the graph being a bowl shape
f. Gradient of f is decreasing. Think about the graph being an upside bowl shape
g. Where graph crosses the x axis
h. Where the graph crosses the y axis
i. Where the gradient of f is zero - the graph is horizontal.
j. Where the gradient of f is positive - the graph is sloping up
The following diagram shows the graph of f', the derivative of f
Fill in the blanks below.
- If more than one answer is required put your answers in alphabetical order
- If an interval is required, put the largest possible interval
a. There are stationary points at and
b. There is a stationary point of inflexion at
c. There is a local minimum at
d. The function is concave down (negative concavity) between and
e. f''(x) = 0 at and
f. There is a non-stationary point of inflexion at
g. The function is increasing in between and
a. stationary points are when f'(x) = 0
b. C is a stationary point of inflexion since at C f'(x)=0. Before and after C the f'(x) < 0.
c. At F, f'(x)=0. Before F, f'(x) < 0 and after F, f'(x) > 0
d. For concave down f'(x) is decreasing
e. This is where the gradient of f'(x) = 0
f. gradient of f'(x) = 0. Before and after this point f'(x) < 0
g. The function is increasing where f'(x) > 0
Let \(f(x)=x^3-6x^2\)
The following diagram shows the graph of f
There are x intercepts at x = 0 and x = c. There is a local minimum at B where x = b and a point of inflexion at A where x = a
a. Find the value of C
b. Find the value of b
c. Find the value of a
d. What is the rate of change of f at x = b
a)
\(f(x)=x^3-6x^2\\ f(x) =x^2(x-6) \)
To find x interceps solve f(x) = 0
x²(x-6)=0
x=0 , x = 6
Hence C = 6
b)
\(f(x)=x^3-6x^2\\ f'(x) =3x^2-12x \)
To find stationary points solve f'(x)=0
3x²-12x = 3x(x-4)=0
x=0 , x=4
Hence b = 4
c)
\(f'(x) =3x^2-12x\\ f''(x) = 6x - 12\)
To find point of inflexion folve f''(x)=0
6x - 12 = 0
x = 2
Since \(f'(2)\neq 0\) there is a non-stationary point of inflexion at x = 2
Hence a = 2
d) The gradient at x = 4 is zero
The graph of y = f(x) is shown below, where B is a local maximum and C is a local minimum
Sketch a graph of y = f'(x), clearly showing the images of the points B and C labellling them B' and C' respectively
Hint
Full Solution
A function is given by \(f(x)=-x^3+6x^2+4\)
a) Find the coordinates of any stationary points and describe their nature
b) Determine the values of x such that f(x) is a increasing function
c) Find the coordinates of the point of inflexion
Hint
Full Solution
The following diagram shows the graph of \(f'\), the derivative of f
On the graph below, sketch the graph of y = f(x) given that f(0) = 0. Mark the images of A , B and C labelling them A' , B' and C'.
Hint
Full Solution
Consider the function \(f(x)=-x^3-3x^2+9x\)
a) Find the coordinates of any stationary points and determine their nature
b) Find the equation of the straight line that passes through both the local maximum and the local minimum points.
c) Show that the point of inflexion lies on this line.
Hint
Full Solution
How much of Graphs and Derivatives have you understood?
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