Maxima & Minima- Using First Derivative Test

Maxima and minima of a function are very important concepts which are applicable both for Mathematics and Science. This poses an obvious question, how to find them?
To start with it, let us see the first method i.e. first derivative test. This method is based on the basic concept of increasing and decreasing functions.

From the definition of the function, we can determine the critical points by f’ = 0. If at any point on the curve where x = a.f'(a) , or is not differentiable at a, then is known as a critical point. If you could find out where the function is increasing and decreasing, we can tell whether the given critical point is a local maximum or minimum. But wait, there is a catch. This first derivative test definition holds true only if the function is continuous at the point x = a, where the test is being applied. If it’s not continuous there, then you can not apply this method. Let us try to understand this with an example.

First Derivative Test For Maxima and Minima

Figure 1: Graph with a local minimum at

In fig. 1, we can see that point

Let us assume a function which has a critical point at

• If
  \(\begin{array}{l} f’ \end{array} \)
  is negative towards just left of
  \(\begin{array}{l} a \end{array} \)
  and it is positive towards just right of
  \(\begin{array}{l} a \end{array} \)
  , then the point
  \(\begin{array}{l} P(x = a)\end{array} \)
  is a local minimum for the function
  \(\begin{array}{l} f\end{array} \)
  .
• If
  \(\begin{array}{l} f’ \end{array} \)
  is positive towards just left of
  \(\begin{array}{l} a \end{array} \)
  and it is negative towards just right of
  \(\begin{array}{l} a \end{array} \)
  , then the point
  \(\begin{array}{l} P(x = a)\end{array} \)
  is a local maximum for the function
  \(\begin{array}{l} f\end{array} \)
  .
• If
  \(\begin{array}{l} f’ \end{array} \)
  has same sign towards just left and just right of
  \(\begin{array}{l} a \end{array} \)
  , then the point
  \(\begin{array}{l} P(x = a)\end{array} \)
  is neither a local maximum nor a local minimum for the function
  \(\begin{array}{l} f\end{array} \)
  .

To summarize what we have learned, let us put it in a tabular form.

Table 1: Different cases possible for first derivative method

Pictorial representation of f’ towards left and right of point P(In all cases, f'(a) = 0)	Nature of f’ Towards just left	Towards just right	Whether Local maximum or minimum
	Negative	Positive	Local Minimum
	Positive	Positive	Neither
	Positive	Negative	Local Maximum

Example

Let us take an example to understand the application of this test. We will try to find out all the critical points for the function and tell whether they are local maxima or minima.

So, our critical points are 0, -2 and -3.

From equation 1,

So, there you go! That was the first derivative test. But as it turns out, you can’t apply this for every function. For example:

‘