How to calculate maximums, minimums, and inflection points in a function

In this lesson I am going to teach you how to calculate maximums, minimums and inflection points of a function when you don’t have its graph.

The relative extremes of a function are maximums, minimums and inflection points (point where the function goes from concave to convex and vice versa).

How to obtain maximums, minimums and inflection points with derivatives

The relative extremes (maxima, minima and inflection points) can be the points that make the first derivative of the function equal to zero:

maximum and minimum inflection points

These points will be the candidates to be a maximum, a minimum, an inflection point, but to do so, they must meet a second condition, which is what I indicate in the next section.

How to know if a point is a maximum, a minimum or an inflection point

Once we have obtained the points for which the first derivative of the function is equal to zero, for each point we must check the following:

If the value of the second derivative at that point is greater than zero, then that point is minimum:

minimum maxima and inflection points

If the value of the second derivative at that point is less than zero, then that point is maximum:

maximos y minimos ejemplos paso a paso

If the second derivative at that point is equal to zero, then that point is an inflection point, as long as the third derivative at that point is other than zero:

minimum maxima and inflection points examples

Let’s see with an example everything explained so far.

Resolved exercise on how to calculate maximums, minimums and inflection points

Let’s get the relative extremes of the next function:

maximum and minimum exercises solved step by step

First, we are going to obtain the possible relative extremes, obtaining the first derivative of the function and equaling it to 0.

The first derivative of the function is:

how to calculate maxima and minima

We equal it to zero to get the points that meet that condition:

steps to calculate maxima and minima

To solve the equation, we previously simplified it:

maximo minimo y puntos de infllexion

As it is a third degree equation, I break it down into factors by the Ruffini rule:

maximos y minimos y puntos de infllexion

Whose solutions are:

calculate maximos y minimos

That correspond to possible maximums, minimums or inflection points.

Now we are going to check what each point corresponds to, studying the sign of the second derivative. For this we obtain the second derivative of the function:

how to calculate inflection points

And we calculate the value of the second derivative for each one of the values that we have just calculated and that make the first derivative zero (x=-2, x=-1 and x=1).

We start by calculating the value of the second derivative for x=-2:

inflection points of a function

The result is greater than zero, so in x=-2 there is a minimum:

maximum and minimum of a function exercises

We calculated the value of f”(x) for x=-1:

inflection points

The result is less than zero, so in x=-1 there is a maximum

tipping point

And finally, we calculate the value of the second derivative for x=1:

inflection points exercises

Whose value is greater than zero, so in x=1 there is a minimum:

inflection point of a function

With the values of x obtained from equalizing the first derivative to zero, we have had no value of f”(x) equal to zero, that is, we have not found any inflection point.

Therefore, we are going to calculate the points that make the second derivative equal to 0:

calculate inflection point

We equate the second derivative to 0:

derived inflection point

And we solve the equation, whose results are:

inflection points calculate

These two values are possible inflection points, as long as they meet the third derivative for those points is other than zero.

We calculate the third derivative of the function:

third derivative

And we find the value of the third derivative for x=0.21:

inflection points

Which is different from 0, so at x=0.21 there is an inflection point:

how to calculate inflection point

We do the same with x=1.24:

derived maxima and minima exercises

The result is also non-zero, so at x=-1,24 there is an inflection point.

How to get inflection points

In short, the relative extremes we have found are:

  • A minimum in x=-2
  • A maximum in x=-1
  • A minimum in x=1
  • An inflection point in x=0.21
  • An inflection point in x=-1,24

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