How to sketch the graph of a differntiable function y = f(x) that has the given properties.Local minimum at (1,1) and local maximum at (3,3)
polynomials are nice differentiable functions.
f'(x) = a(x-1)(x-3)
f(x) = a/3 x(x-3)^2 + b
f(3) = 3 = b
f(1) = 1 = 4a/3 + b = 4a/3+3, so a = -1/2
f(x) = -1/2 x(x-3)^2 + 3
To sketch the graph of a differentiable function with the given properties, follow these steps:
1. Start by plotting the local minimum at (1, 1) and local maximum at (3, 3) on the coordinate plane.
2. Since the given function is differentiable, the graph will have a continuous slope. This means that between (1, 1) and (3, 3), there will be a point where the slope is zero. This point will be either a local maximum or a local minimum.
3. To find this point, set the derivative of the function, denoted as f'(x), equal to zero and solve for x. The resulting x-value will give you the x-coordinate of the point where the slope is zero.
4. After obtaining the x-coordinate, substitute it back into the original function to find the corresponding y-coordinate.
5. Plot the point obtained in step 4 on the graph.
6. The points (1, 1), (3, 3), and the point obtained in step 4 will form a general shape or curve. Observe the direction of the slope on either side of these points and sketch the graph accordingly.
7. Consider the behavior of the function for large values of x (both positive and negative) to determine the overall shape of the graph. For example, if the function approaches positive or negative infinity as x approaches positive or negative infinity, respectively, the graph may have an upward or downward trend.
8. Connect the plotted points and shape the graph smoothly based on the information gathered from steps 6 and 7.
Remember that these steps provide a general guideline and may vary depending on the specific properties and characteristics of the function.