Consider the following graph of the function g. From the given graph of g, state the numbers at which g is discontinuous. (Enter your answers as a comma-separated list.)
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.
![### Analyzing Discontinuities in the Graph of Function \( g \)
#### Task
Consider the following graph of the function \( g \).
#### Graph Description
The graph depicts the function \( g \) plotted on the Cartesian plane. Key observations from the graph include:
- The x-axis ranges from approximately -3 to 4, and the y-axis spans from -3 to 3.
- The curve exhibits several distinct features, including breaks (discontinuities).
#### Points Noted
From the graph, identify the x-values at which \( g \) is discontinuous.
#### Graph Breakdown
1. **Near \( x = -1 \)**:
- The curve shows a break, possibly indicating a point of discontinuity.
2. **At \( x = 0 \)**:
- There's a vertical asymptote, signifying the function is undefined and discontinuous at this point.
3. **Near \( x = 1 \)**:
- The function exhibits a removable discontinuity, represented by a hole in the graph.
4. **At \( x = 2 \)**:
- Similar to \( x = 1 \), there’s a clear interruption in the plot, indicating another point where \( g \) is discontinuous.
#### Instructions
From the given graph of \( g \), state the x-values where \( g \) is discontinuous. Enter your answers as a comma-separated list in the form field provided.
#### Input Field
- **x =** \[________\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb8a73d8c-64dc-4e32-8296-8ed66dd27ec0%2F58d5c151-0ade-47e2-bf2e-08adae82f231%2F7plpj_processed.png&w=3840&q=75)
![# Limit and Continuity of Piecewise Functions
## Sketch the Graph of the Function:
Given the piecewise function:
\[
f(x) =
\begin{cases}
\sqrt{x} & \text{if } x \leq -1 \\
x & \text{if } -1 < x \leq 2 \\
(x-1)^2 & \text{if } x > 2
\end{cases}
\]
## Graph Explanation:
The image presents four different graphs. Each graph attempts to sketch the function \( f(x) \) given above.
### Graph Analysis:
1. **Graph (a)**:
- For \( x \leq -1 \): The function is \( \sqrt{x} \) and the curve starts from the negative \( x \)-axis moving upward.
- For \( -1 < x \leq 2 \): The function changes to \( x \), depicted as a straight line passing through the origin.
- For \( x > 2 \): The function is \( (x-1)^2 \), shown as a parabola.
- The graph displays a smooth transition between these segments with no apparent breaks.
2. **Graph (b)**:
- The description for \( x \leq -1 \), \( -1 < x \leq 2 \), and \( x > 2 \) segments is similar in all the graphs.
- However, graph (b) places a hollow point at \( x = -1 \) and \( x = 2 \), indicating discontinuities.
3. **Graph (c)**:
- This graph highlights a solid point at the transition at \( x = -1 \) and a hollow point at \( x = 2 \).
4. **Graph (d)**:
- In graph (d), for \( x > 2 \), a hollow point is evident while for \( -1 < x \leq 2 \) segment, there is a solid point shown at \( x = 2 \).
## Interval Notation:
The question below the graphs asks:
"Use the graph to determine the values of \( a \) for which \(\lim_{x \to a} f(x)\) exists. (Enter your answer using interval notation.)"
Given the correct graph selection (graph d), the appropriate interval notation for the values of](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb8a73d8c-64dc-4e32-8296-8ed66dd27ec0%2F58d5c151-0ade-47e2-bf2e-08adae82f231%2Fs6dodyh_processed.png&w=3840&q=75)
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