Hopefull the last one. :)
Sketch the graph of a differentiable function y = f(x) with this property:
A local minimum value that is greater than one of its local maximum values
To sketch the graph of a differentiable function with a local minimum value greater than one of its local maximum values, we need to understand the concepts of local minimum and maximum values.
A local minimum occurs at a point where the function reaches the lowest value in a specific interval and the function is increasing both to the left and right of that point. Conversely, a local maximum occurs at a point where the function reaches the highest value in a specific interval and the function is decreasing on both sides of that point.
Here's a step-by-step approach to sketching such a graph:
1. Start by considering a basic function that has a local minimum and a local maximum. For example, let's take the function y = x^3.
2. Plot the graph of y = x^3 on a coordinate plane. This graph will have a local minimum at the origin (0,0) and a local maximum at (1,1).
3. Next, modify the function to shift the local minimum point upward. To accomplish this, add a constant value to the expression. For instance, let's consider the function y = x^3 + 2.
4. Plot the graph of y = x^3 + 2 on the same coordinate plane. Now the local minimum will be at (0,2), which is higher than the local maximum at (1,1).
5. Finally, connect the points on the graph smoothly, ensuring that the function maintains its differentiability. The graph should have a smooth curve passing through the local minimum and maximum points.
In summary, to sketch the graph of a differentiable function with a local minimum value that is greater than one of its local maximum values, modify a basic function with a local minimum and maximum by adding a constant to shift the local minimum point upwards.