First year:
(I)
Tangent, maximum and minimum
Tangents
The tangent to the graph of a function f at the point c, f (c) is a line such
that:
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its slope is equal to f ' (c).
it passes through the point c, f (c).
The equation of the tangent to the graph of a function f at the point c, f (c)
is given by the following formula:
y  f ' ( x)( x  c)  f ( x).
Example: Find the equation of the tangent to the graph of f ( x)  x 2 at the
point (1,1).
We have f ' ( x)  2 x and, since c  1 , we obtain
y  f ' (1)( x  1)  f (1) 
y  2( x  1)  1 
y  2 x  1.
(II)
Maximum and minimum
A function f (x) is said to have a local maximum at x0 if there exists a  0
such that, for x  ( x0  a, x0  a) , we have f ( x)  f ( x0 ) .
Intuitively, it means that around x0 the graph of f will be below f ( x0 ) .
Similarly, a function f (x) is said to have a local minimum at x0 if there
exists a  0 such that, for x  ( x0  a, x0  a) , we have f ( x)  f ( x0 ) .
This time, the graph of f will be situated above f ( x0 ) for values of x around
x0 .
Examples:
f ( x)  x 2  x  3 .
From the graph, it is rather obvious that the function has a unique minimum
and that this minimum is global (i.e. the whole graph is above this minimum).
On the other hand, if we take f ( x)  x 3  4 x 2  3x  2 , the situation is rather
different:
Here, we have a local maximum and a local minimum.
Minima and maxima have one thing in common: say f has a local minimum
at x0 . Then the tangent to the graph of f at the point x0 , f ( x0 )  is a
horizontal line:
The slope of the tangent is therefore 0 .
Remember, the slope of the tangent to the graph of f at the point x0 , f ( x0 ) 
is equal to f ' ( x0 ), so here we end up with f ' ( x0 )  0 .
If f has a local minimum or a local maximum at x0 , we therefore have
f ' ( x0 )  0 .
In general, the solutions of f ' ( x)  0 are called stationary points. There are
three different kinds of stationary points: local minima, local maxima and
turning points.
You can classify them as follows:
Say x0 is a stationary point. Then if
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f ' ' ( x0 )  0 , there is a local maximum at x0 .
f ' ' ( x0 )  0 , there is a local minimum at x0 .
f ' ' ( x0 )  0 , there is a turning point at x0 .
Example: f ( x) 
x3 x 2
  6 x  2 . Find and classify the stationary points of f .
3
2
To find the stationary points, we solve f ' ( x)  0 :
Here, f ' ( x)  x 2  x  6  ( x  2)( x  3) , so that f ' ( x)  0 
x  2 or
x  3 .
Next, we calculate f ' ' ( x) and use the rule above to classify the stationary
points:
f ' ' ( x)  2 x  1 .
f ' ' (2)  5  0 , so that f has a local minimum at x  2.
f ' ' (3)  5  0 , so that f has a local maximum at x  3 .
Let’s have a look at the graph of f :
The graph indicates that there is indeed a local minimum at x  2 and a local
maximum at x  3 . The graph also indicates that they are both local and not
global.