Maxima and the Minima of Functions

Extrema are the maximum and minimum values of a function. Maxima and minima are the highest and lowest values of a function within the defined ranges. The largest value of the function is known as the absolute maximum, and the minimum value is known as the absolute minimum for the complete range of the function.

Using Derivatives to Discover the Maximum and Minimum

The concept of derivatives is used to identify maxima and minima. As we are aware that the concept of derivatives provides information about the gradient or slope of a function, we locate the places where the gradient is zero, which we refer to as turning points or stationary points. These are the points associated with the greatest or smallest values of the function (locally).

Derivative Tests

The derivative test is used to determine the maximums and minimums of any function. In general, first- and second-order derivative tests are utilised. Let’s examine this in-depth.

First Order Derivative Test:

Consider f to be the open interval I’s defined function. Additionally, f must be continuous at the crucial point c in I so that f'(c) = 0.

If f'(x) becomes negative as x increases through point c, then point c is the local maximum, and f(c) is the maximum value.

If f'(x) becomes positive as x increases through point c, then point c is the local minimum, and f(c) is the minimum value.

If the sign of f'(x) does not change as x increases through c, then c is neither a local minimum nor a local maximum.

Second-Order Derivative Test:

Consider f to be the interval I-defined function that is twice differentiable at c.

If f'(c) = 0 and f”(c) < 0, x = c will be the location of local maximum if x = c. Consequently, f(c) will have a local maximum value.

If x = c, the point of local minimum will be there. f'(c = 0 and f”(c) < 0 if f'(c = 0 and f”(c) < 0. Consequently, f(c) will have a local minimum value.

When both f'(c) and f”(c) = 0, the test fails. And this first derivative test will provide us with the local maximum and minimum values.

Application of Partial Derivative

If they exist, partial derivatives can be utilised to determine the maximum and minimum value of a two-variable function. We attempt to locate a stationary point with zero slopes and then trace the surrounding maximum and minimum values. The practical use of maxima and minima is to maximise profit or minimise losses for a given curve.

Suppose f(x,y) is a real-valued function and (pt,pt’) is the interior points of its domain.

• pt, pt’ is referred to as a point of local maximum if there exists a h > 0 such that f(pt,pt’) f(x,y) for all x in (pt – h, pt’ + h) and xa. The value f(pt,pt’) is referred to as f’s local maximum value (x,y).

• pt, pt’ is a point of local minima if there exists a < h < 0 such that f(pt,pt’) f(x,y) for any x in (pt – h, pt’ + h) and x a. f(pt,pt’) is known as the local minimum value of f. (x,y).

Algorithm for locating the maxima and minima of functions with two variables:

1. Find the values of x and y using the equation f_xx= 0 and  f_yy=0 f_xx and f_yy are the function’s partial double derivatives concerning x and y, respectively.

2. The obtained result will be regarded as the curve’s stationary/turning points.

3. Create three new variables named r, t, and s.

4. Determine r, t, and s using r=f_xx, t= f_yy, and s=f_xy

5. If (rt-s2)|(stationary pts)>0 (Maximum/Minimum), the condition is met.

6. If (rt-s2)|(stationary points) 0 (No Maximum/Minimum), then (Saddle point)

7. If r = f_xx > 0 (Minima)

8. If r = f_xx < 0 (Maxima),

The characteristics of maximums and minimums are as follows:

If f(x) is a continuous function in its domain, then at least one maximum or minimum must lie between values of f that are equal (x).

The maximum and minimum values alternate. Thus, there is one minimum between each pair of maxima and vice versa.

f(c) is the minimal and least value if f(x) tends to infinity as x approaches an or b and f'(x) = 0 for only one value x, namely c, between a and b. If f(x) approaches – ∞ as x approaches an or b, then f(c) is the maximum and greatest value.

Notes on the Maxima and Minimum:

• Maximums and minimums are the peaks and valleys of a function’s curve.

• On the entire domain, there can be only one absolute maximum and one absolute minimum of a function.

• A purpose A function is said to be monotonous in the interval I if it either increases or decreases in I.

Conclusion

Maximums and minimums are the peaks and valleys of a function’s curve. A function can have any number of maxima and minima. Calculus allows us to determine the maximum and minimum value of any function without examining its graph. The maxima will be the highest point on the curve within the specified range, while the minima will be the lowest point.