S(r) = In (x² + 2x + 5)
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.
Find the points of inflection and determine the intervals on which f(x) is concave up and down.
![The function displayed in the image is:
\[ f(x) = \ln(x^2 + 2x + 5) \]
This is a logarithmic function where the natural logarithm (denoted by \(\ln\)) is applied to a quadratic expression \(x^2 + 2x + 5\).
**Explanation for Educational Purposes:**
1. **Function Definition:**
- \( f(x) \): The notation indicates that \(f\) is a function of \(x\).
- \(\ln(\cdot)\): This denotes the natural logarithm, which is the logarithm to the base \(e\) (where \(e \approx 2.718\)).
2. **Inside the Logarithm:**
- \( x^2 \): This is the square of \(x\).
- \( 2x \): This represents twice the value of \(x\).
- \( 5 \): This is a constant.
3. **Quadratic Expression:**
- \( x^2 + 2x + 5 \): This quadratic expression is the argument of the natural logarithm function.
4. **Domain of the Function:**
- For the natural logarithm function to be defined, the argument (inside the logarithm) must be greater than 0.
- Thus, \( x^2 + 2x + 5 > 0 \) must hold true for the function to be defined.
**Graph Description:**
- To visualize this function, one would plot the quadratic expression \(x^2 + 2x + 5\) first.
- The graph of \(f(x)\) would then reflect how the natural logarithm of these values behaves, typically showing an increasing trend where the logarithm function is defined.
This representation helps in understanding how the function behaves and what its domain and range might be in a graphical context.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F08a51670-5bf9-44d1-ad0c-251791c78189%2F2943a3b1-a993-4c77-bfef-f296f563edaa%2F4vlzibp_processed.png&w=3840&q=75)
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