Determine where the function is concave upward and where it is concave downward. g (x) = 5+ 22 Concave upward: 0(-00-V15), (0, V15) 0l-v15,0), (/15, 0) 0l-VI5, VT5) o(-00,-VI5), (VT5, ∞0) O no interval Concave downward: ol-00,-V15), (0, V15) o-V15,0), (VT5, ∞) ol-VI5, V15) o-00,-VT5). (VT15, 0) O no interval
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.
![**Determining Concavity of the Function**
To determine where the function \( g(x) = \frac{x}{5 + x^2} \) is concave upward or concave downward, follow the steps below.
### Function:
\[ g(x) = \frac{x}{5 + x^2} \]
### Concave Upward:
- \((-\infty, -\sqrt{15})\), \( (0, \sqrt{15}) \)
- \((-\sqrt{15}, 0)\), \( (\sqrt{15}, \infty) \)
- \((-\sqrt{15}, \sqrt{15}) \)
- \((-\infty, -\sqrt{15})\), \( (\sqrt{15}, \infty) \)
- No interval
### Concave Downward:
- \((-\infty, -\sqrt{15})\), \( (0, \sqrt{15}) \)
- \((-\sqrt{15}, 0)\), \( (\sqrt{15}, \infty) \)
- \((-\sqrt{15}, \sqrt{15}) \)
- \((-\infty, -\sqrt{15})\), \( (\sqrt{15}, \infty) \)
- No interval
### Explanation:
To determine concavity, you need to find the second derivative of \( g(x) \). Using the second derivative test, analyze the sign of the derivative:
1. Find the second derivative, \( g''(x) \).
2. Determine the intervals on which \( g''(x) \) is positive (concave upward) and where it is negative (concave downward).
3. Identify the points where \( g''(x) = 0 \) or does not exist and use these points to define the intervals.
Identify each interval and verify whether it corresponds to concave upward or downward behavior of the function, based on where the second derivative is positive or negative. The given options aim to guide you in identifying these intervals correctly.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F376a455d-f759-4362-ac1e-6344c6be0004%2F82738a5b-c247-4e86-a584-562aa123a46b%2Fog5k0e_processed.jpeg&w=3840&q=75)
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