Applied Calculus: UNIT I: Differential Calculus
Local maxima and local minima of functions of single variable
Differential Calculus
Explanation, Formula, Equation, Example - Differential Calculus: Local maxima and local minima of functions of single variable
LOCAL MAXIMA AND LOCAL MINIMA
Consider the following graph
In the interval [a, b], f(x) increasing function. In the interval [b, c], f(x) is decreasing function. In the interval [c, d], f(x) is increasing function.
The graph shown in Fig. rises from A to B, falls from B to C, and rises again from C to D. The function f is said to be increasing on the interval [a, b], decreasing on [b, c], and increasing again on [c, d]. Notice that if x1 and x2 are any two numbers between a and b with x1 < x2, then f(x1) < ƒ(x2).
From the above concept, we can define the increasing and decreasing functions as given below:
Increasing function
A function f is called increasing function on an interval I, if f(x1) < f(x2) where as x1 < x2
Decreasing function
A function ƒ is called decreasing function on an interval I, if f(x1) > f(x2) where as x1 < x2
Example:
Consider the following example f(x) = x2 for x ∈ R.
In the interval (‒∞, 0], the function is decreasing and in the interval [0, ∞), the function is increasing.
Let c be that point in a domain D on a function f, then f(c) is
• local maximum value of f, if f(c) ≥ f(x) when x is near c.
• local minimum value of f, if f(c) ≤ f(x) when x is near c.
Consider the following figure.
The absolute minimum = f(a)
The absolute maximum= f(d)
f(x) is increasing in [a, b]
f(x) is decreasing in [b, c]
f(x) is increasing in [c, d]
f(x) is decreasing in [d, e]
In [a, c], the local minimum is f (a) and the local maximum is f (b)
In [d, e], the local maximum is f(d) and the local minimum is f(e)
The following figure shows local and absolute maximums occur at a single point and similarly shows the local and absolute minimums occur at a single point.
Applied Calculus: UNIT I: Differential Calculus : Tag: Applied Calculus : Differential Calculus - Local maxima and local minima of functions of single variable
Applied Calculus: UNIT I: Differential Calculus
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Applied Calculus
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