Monotony . It can be characterized as the movement of our eyes when we run the graph of the function from left to right. Its evaluation allows determining (in the graph) points known as extremes, where the function changes its monotony, has stopped growing and is not yet decreasing, or vice versa. .
The monotony can be analyzed both intervals to specific points of a given function.
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- 1 monotony on and off
- 1 Monotony of functions in a branch
- 1.1 Monotony at one point
- 2 Monotony and extremes according to criteria of the first and second derivatives
- 3 Physical meaning of monotony
- 4 Calculation of extremes. Optimization problems
- 1 Monotony of functions in a branch
- 2 Sources
Monotony on and off
By analyzing real functions in one variable, up to five different categories can be defined. .
Strictly increasing function in an interval.
1. The function is said to be strictly increasing on the interval (a, b) when for all x1; x2 ∈ (a, b), such that x1 <x2 holds that f (x1) <f (x2). This graphically means, as noted above, that by moving right on it the function only goes up. .
2. Increasing function in an interval.
The function is said to be increasing on the interval (a, b) when for all x1; x2 ∈ (a, b), such that x1 <x2 is true that f (x1) ≤f (x2). In this case it is necessary to point out that both conditions (equality and growth) are verified throughout the interval. This means graphically, as noted above, that as you move to the right on it the function is constant and goes up ..
3. Constant function on an interval
The function is said to be constant on the interval (a, b) when for all x1; x2 ∈ (a, b), such that x1 <x2 is true that f (x1) = f (x2). This graphically means, as noted above, that as you move to the right on it the function is constant. The functions of the form f (x) = k are constant throughout their domain.
4. Decreasing function on an interval
The function is said to be decreasing on the interval (a, b) when for all x1; x2 ∈ (a, b), such that x1 <x2 holds that f (x1) ≥f (x2). In this case, it is necessary to point out that both conditions (equality and decrease) are verified throughout the interval. This means graphically, as noted above, that by moving to the right on it the function is constant or low. 5. Strictly decreasing function on an interval
The function is said to be strictly increasing in the interval (a, b) when for all x1; x2 ∈ (a, b), such that x1 <x2 holds that f (x1)> f (x2). . This graphically means, as noted above, that by advancing to the right on it the function only drops. Some authors prefer to use for the categories 2 and 4 the terms “not decreasing” and “not increasing”, taking into account that if the function “rises or is constant” then “does not decrease” while if “falls or is constant” then does not grow. In this case, for categories 1 and 5, it would suffice to use the terms increasing and decreasing respectively.
Monotony of functions in a branch
For the functions defined in a branch, it is impossible to verify the ¨intermediate¨ categories, that is, 2 and 4. Given a set of ordered pairs (x; f (x)) for all x∈ R, the graph can grow and decrease but it cannot be constant and grow or decrease. For functions defined in this way, the use of the monotony criterion at one point is very practical.
To analyze the monotony of the function at a given point we will analyze the behavior in the vicinity of the point.
Monotony at one point
A function f (x) is increasing at a point X0 if for all ε infinitely small and positive, it is true that f (X0-ε) <f (X0) <f (X0 + ε), similarly a function f (x) it is decreasing at a point X0 if for all ε infinitely small and positive, it is true that f (X0-ε)> f (X0)> f (X0 + ε).
We will then say that a function is increasing (decreasing) on the interval (a, b) if it is increasing (decreasing) for all x∈ (a, b).
Extreme points. They are the highest and lowest points on the graph of a function. At these points, this function neither increases nor decreases. They can be of two types:
- a) Relative maximum: in an environment close to the point on the left the function increases and in an environment on the right the function decreases, there is a monotony change from increasing to decreasing: f (x0)> f (x0-) and f ( x0)> f (x0 +)
- b) Relative minimum: in an environment close to the point on the left the function decreases and in an environment on the right the function increases, there is a monotony change from decreasing to decreasing: f (x0) <f (x0-) and f ( x0) <f (x0 +)
If the maximum (minimum) is for the entire function it is said absolute. If the maximum (minimum) is only for one zone of the function, it is called local.
Monotony and extremes according to criteria of the first and second derivatives
- If a function is differentiable, then the first derivative evaluated at a point is equal to the slope (m) of the tangent line to the curve at that point.
- If the angle that forms a line with the positive semi-axis of the abscissa is: 0º then the slope (m) is equal to 0
Sharp then the slope (m) is positive.
Obtuse then the slope (m) is negative
Let f be a differentiable function on the interval (a, b).
- If f ‘(x)> 0 for all x of (a, b), then f is increasing by (a, b).
- If f ‘(x) <0 for all x of (a, b), then f is decreasing in (a, b).
- If f ‘(x) = 0 for all x of (a, b), then f is constant in (a, b).
Theorem 2: Let f be a function with its first derivative defined, at least, on an open interval containing the number a. If f´´ is defined then it is true:
- a) .- If f´ (a) = 0 and f´´ (a) <0 then f is said to have a local maximum in a.
- b) .- If f´ (a) = 0 and f´´ (a)> 0 then f is said to have a local minimum in a.
To see how to apply the above theorems, note that for continuous f, f ‘(x) only changes sign around its zeros, then to determine the intervals in which f is increasing or decreasing, we suggest the following steps:
- Locate the intercepts with the “x” axis of the first derivative of f. (solving the equation f ‘(x) = 0)
- Determine the sign of f ‘(x) at a point in each interval determined by two consecutive critical points.
- Decide, using Theorem 1, whether f is increasing or decreasing in each of these intervals.
- Classify the extreme points using Theorem 2.
Physical meaning of monotony
When analyzing the behavior of a vector physical magnitude (which has direction and sense) over time in a system of coordinate axes where we place the physical magnitude (which we will call primary) on the axis of the abscissa and the time , the determination of the monotony intervals and extreme points of said graph allows us to know the meaning of another physical quantity (which we will call a derivative). Thus, for example, in projectile launches, the evaluation of the monotony of a height versus time graph allows us to know the direction of the vertical speed of the projectile; analogously, the evaluation of this property in a graph of electric charge vs. time, allows us to characterize the direction of the electric current; and in the case of magnetic flux vs. time, to the induced electromotive force.
Let’s look at the first example:
The height in meters of an object thrown from the ground upwards after t seconds is given by the equation h (t) = 96t − 16t2.
Calculating the zeros of this inverted parabola (0 and 6), we know that the flight time is 6 seconds and since the quadratic function is symmetric with respect to the line x = xv, the vertex abscissa is x = 3 and its ordinate f ( 3) = 144. The vertex constitutes a global maximum of this function. The monotony intervals are: (0,3) increasing; (3,6) decreasing, at x = 3 there is a maximum, therefore the function changes the monotony from increasing to decreasing, let’s analyze the physical meaning of the monotony of h vs t, in the instantaneous variation of h, that is, the vertical component of the velocity of the object. In the increasing section (0,3), vy is directed upwards, the object is rising (figure 4a); in the decreasing section (3,6), vy is directed downwards (figure 4c), the object is descending; while at the local maximum,
Extremes calculation. Optimization problems
Sometimes we are interested in solving situations in which it is necessary to find a value that makes another maximum or minimum, these problems are called “optimization”. Optimization problems come down to getting the relative extremes of a function.
These problems often require a prior approach, which, in summary, is as follows:
- Determine the function from which you want to obtain the maximum or minimum. It is easy for it to depend on more than one variable; In this case, look for the relationship between them so that we only have one unknown. • Calculate the maximum or minimum order, imposing the necessary conditions on its derivatives
MAXIMUM f ‘(x0) = 0 and f (x0) <0
MINIMUM f ‘(x0) = 0 and f (x0)> 0