Sketch a graph of a function f(x) that is differentiable and that satisfies the following conditions.
c) f'(-3) = 0 and f'(1) = 0
Please show me, step by step, how to sketch the problem!
There is more than one solution to this problem. Try an polynomial equation of with a derivative that must satsify the two zero points:
f'(x) = (x+3)(x-1) = x^2 +2x -3
f(x) = x^3/3 + x^2 -3x + (any constant)
Note that there is an arbitrary constant, so there are already an infinite number of functions. I could just as well have picked a cosine function f(x) = cos (a*pi*x) with zero slope
f'(x) = - a pi sin a*pi*x = 0
at integer values of x, with a being any integer
(ax = n*pi)
We can't draw graphs for you. You can do that.
To sketch a function that satisfies the given conditions, we can follow these steps:
Step 1: Use the fact that f'(-3) = 0 to determine whether there is a local extremum (maximum or minimum) at x = -3. Since the derivative is zero at x = -3, it suggests that the graph could have a horizontal tangent at that point. However, we need more information to determine if there is a local extremum or not.
Step 2: Use the fact that f'(1) = 0 to determine whether there is a local extremum at x = 1. Similar to the previous step, the derivative being zero at x = 1 indicates a potential horizontal tangent. We also need more information to determine if there is a local extremum at this point.
Step 3: Determine the nature of the function before and after x = -3 and x = 1. This will help us understand the behavior of the graph and its possible shape.
To proceed with the sketch, we will consider a few cases:
Case 1: Both x = -3 and x = 1 are local maxima or minima:
- In this case, the graph will have a horizontal tangent at x = -3 and x = 1.
- To determine the shape of the graph, we need additional information, such as the concavity of the function or information about the second derivative.
Case 2: Only x = -3 is a local maximum or minimum:
- In this case, the graph will have a horizontal tangent at x = -3 but not at x = 1.
- The shape of the graph could be ascending before x = -3, and descending after x = -3, or vice versa.
Case 3: Only x = 1 is a local maximum or minimum:
- In this case, the graph will have a horizontal tangent at x = 1 but not at x = -3.
- The shape of the graph could be ascending before x = 1, and descending after x = 1, or vice versa.
Case 4: Neither x = -3 nor x = 1 is a local maximum or minimum:
- In this case, the graph may still have horizontal tangents at x = -3 and x = 1, but they won't be local maxima or minima.
- The shape of the graph could be either continuously ascending or descending.
Unfortunately, without additional information or constraints, it is challenging to determine the exact shape of the graph that satisfies the given conditions. However, by considering the possible cases and understanding the behavior of the function in the vicinity of x = -3 and x = 1, you can make informed decisions while sketching the graph.