Use the function h(x) = x + x2 - z to complete the problems given below. a) The function h(x) has two critical values A and B, with A < B. Identify those two values below: A = В - b) Use the First Derivative Test to determine what kind of local extrema (local minimum or local maximum) the function has at each of these values. Show your work by creating a Sign Chart (a number line that includes the critical values and the sign of the derivative in the intervals between the critical values) and uploading an image of the Sign Chart in the box below. U x, x' A <> Edit - Insert Formats - c) Based on your work in part (b), identify the type of local extrema that exist at each critical value from part (a). Critical Value A O Local Minimum O Local Maximum ONeither Critical Value B O Local Minimum OLocal Maximum O Neither
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.
![**Critical Points and Extrema Analysis Using the First Derivative Test**
Use the function \( h(x) = x + x^2 - x^3 \) to complete the problems given below.
**a)** The function \( h(x) \) has two critical values \( A \) and \( B \), with \( A < B \). Identify those two values below:
```
A =
B =
```
**b)** Use the First Derivative Test to determine what kind of local extrema (local minimum or local maximum) the function has at each of these values. Show your work by creating a Sign Chart (a number line that includes the critical values and the sign of the derivative in the intervals between the critical values) and uploading an image of the Sign Chart in the box below.
```
[Insert Sign Chart Here]
```
**c)** Based on your work in part (b), identify the type of local extrema that exist at each critical value from part (a).
```
Critical Value \( A \)
☐ Local Minimum
☐ Local Maximum
☐ Neither
```
```
Critical Value \( B \)
☐ Local Minimum
☐ Local Maximum
☐ Neither
```](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb0d58f0b-90fc-4a9f-a40d-3617c86d0254%2Fe1b3ec52-f2b1-4848-97cc-99eaea11f7d7%2F6q9v43u_processed.jpeg&w=3840&q=75)
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