For which values of x is f(x) neither decreasing nor increasing?
Understand the Problem
The question is asking for the values of x where the function f(x) is neither decreasing nor increasing, implying that we need to identify the stationary points on the graph provided.
Answer
The values of $x$ for stationary points are found where $f'(x) = 0$.
Answer for screen readers
The stationary points are the values of $x$ for which $f'(x) = 0$. Without the specific function provided, I cannot calculate the exact values.
Steps to Solve
- Identify the function
First, ensure you have the function $f(x)$ defined. For stationary points, you generally need its derivative.
- Find the derivative
Calculate the derivative of the function $f(x)$ to determine where the slope of the tangent (the rate of change) is zero. You will be looking for points where $f'(x) = 0$.
- Set the derivative to zero
Solve the equation $f'(x) = 0$. This will yield the stationary points, which are the x-values where the function is neither increasing nor decreasing.
- Analyze the second derivative (optional)
If desired, you can calculate the second derivative, $f''(x)$, to determine the nature (max/min or inflection) of the stationary points found.
- Collect the stationary points
Compile the values of $x$ from your solutions into a list of stationary points.
The stationary points are the values of $x$ for which $f'(x) = 0$. Without the specific function provided, I cannot calculate the exact values.
More Information
Stationary points are important in calculus as they indicate local maxima, minima, or points of inflection on the graph of a function. Identifying these points helps in understanding the behavior of the function.
Tips
- Not setting the derivative correctly to zero.
- Forgetting to test if the points found are indeed stationary by verifying if they impact the slope (sometimes need second derivative test).
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