Minimization
In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in terms of global or local range. The definition of minimization in the thesaurus is the process of reducing something to a small amount, value, or position. Minimization (noun) is an instance of belittling or disparagement.
Maxima and Minima
The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.
Derivatives
A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which one would be easier? Walking on the road would be much easier than climbing a mountain.
Concavity
In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the first derivative test and second derivative test to understand the concave behavior of the function.
![EXAMPLE 1
Find where the function f(x) = 3x4 – 16x3 – 270x2 + 7 is increasing and where it is
-5
10
decreasing.
- 2000
12:(x -O)(*+D)
-4000
SOLUTION
f'(x) = 12x3 – 48x² – 540x =
- 6000
-8000
To use the I/D Test, we have to know where f'(x) > 0 and where f'(x) < 0. This depends on the signs of
the three factors of f'(x), namely, 12x, x –
-10 000
and x +
We divide the real line into intervals
-12 000
whose endpoints are the critical numbers
(smallest), 0 and
(largest) and arrange our work in a
-14 000
chart. A plus sign indicates that the given expression is positive, and a negative sign indicates that it is
negative. The last column of the chart gives the conclusion based on the I/D Test. For instance, f '(x) < 0 for
0 < x < 9, so f is ---Select--- v on (0, 9). (It would also be true to say that f is decreasing on the closed
interval [0, 9].)
Video Example
Interval
12x
х— 9
X + 5
f'(x)
f
х< -5
decreasing on (-∞, -5)
-5 < x < 0
---Select--- ♥
on (-5, 0)
+
+
0 < x < 9
+
decreasing on (0, 9)
x > 9
+
+
+
+
V ---Select---
(9, о»)
decreasing
increasing
The graph of f shown in the figure confirms L.
the chart.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F982543f2-46cf-4f2e-9047-56f1e64038c1%2Ff2168f1c-a26b-45f8-89e9-dc91208385a8%2Fsu522t_processed.png&w=3840&q=75)
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