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.
![### Calculus: Finding the Derivative
**Problem Statement:**
Find \( f'(x) \) if \( f(x) = e^{3x} \sin 2x \).
**Explanation:**
To find the derivative \( f'(x) \) of the given function \( f(x) = e^{3x} \sin 2x \), we need to use the product rule. The product rule states that if \( f(x) = u(x)v(x) \), then
\[
f'(x) = u'(x)v(x) + u(x)v'(x)
\]
In this case, let:
- \( u(x) = e^{3x} \)
- \( v(x) = \sin 2x \)
We first find the derivatives \( u'(x) \) and \( v'(x) \):
1. \( u(x) = e^{3x} \):
\[
u'(x) = 3e^{3x} \quad \text{(using the chain rule)}
\]
2. \( v(x) = \sin 2x \):
\[
v'(x) = 2\cos 2x \quad \text{(using the chain rule)}
\]
Now we apply the product rule:
\[
f'(x) = u'(x) v(x) + u(x) v'(x)
\]
Substituting the values:
\[
f'(x) = (3e^{3x}) (\sin 2x) + (e^{3x}) (2\cos 2x)
\]
Simplifying:
\[
f'(x) = 3e^{3x} \sin 2x + 2e^{3x} \cos 2x
\]
Therefore, the derivative \( f'(x) \) is:
\[
f'(x) = 3e^{3x} \sin 2x + 2e^{3x} \cos 2x
\]
This is the required derivative of the given function \( f(x) \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fedc85e3b-dc2d-4f8b-a097-9d7c53c76aad%2Fbadfff23-d942-4b11-bc1d-54a6ea4b89a4%2F6vrtpq.jpeg&w=3840&q=75)
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 2 images
