Sketch the graph of a differentiable function y = f(x) with the following properties.
f(2) = 3, f'(2) = 0, and
(a) f'(x) > 0 for x < 2, f'(x) < 0 for x > 2
(b) f'(x) < 0 for x < 2, f'(x) > 0 for x > 2
(c) f'(x) < 0 for x not equalt to 2
(d) f'(x) > 0 for x not equal to 2
Please help.
To sketch the graph of a differentiable function with the given properties, we can start by considering the information about the function's value and its derivative at x = 2.
(a) For f'(x) > 0 for x < 2 and f'(x) < 0 for x > 2, this indicates that the function is increasing for x < 2 and decreasing for x > 2. Since f'(2) = 0, this suggests that the graph has a local maximum or minimum at x = 2.
To sketch the graph, start by marking a point at (2, 3). Since f'(x) > 0 for x < 2, the function is increasing to the left of x = 2. Therefore, draw a curve that slopes upward to the left of (2, 3). Similarly, since f'(x) < 0 for x > 2, the function is decreasing to the right of x = 2. Thus, draw a curve that slopes downward to the right of (2, 3).
(b) For f'(x) < 0 for x < 2 and f'(x) > 0 for x > 2, this indicates that the function is decreasing for x < 2 and increasing for x > 2. Again, since f'(2) = 0, the graph will have a local maximum or minimum at x = 2.
Start by marking a point at (2, 3). Since f'(x) < 0 for x < 2, the function is decreasing to the left of x = 2. So, draw a curve that slopes downward to the left of (2, 3). Similarly, since f'(x) > 0 for x > 2, draw a curve that slopes upward to the right of (2, 3).
(c) For f'(x) < 0 for x not equal to 2, this means the function is always decreasing, except possibly at x = 2.
Start by marking a point at (2, 3). Since f'(x) < 0 for x not equal to 2, draw a curve that slopes downward throughout the entire graph.
(d) For f'(x) > 0 for x not equal to 2, this means the function is always increasing, except possibly at x = 2.
Start by marking a point at (2, 3). Since f'(x) > 0 for x not equal to 2, draw a curve that slopes upward throughout the entire graph.
Remember, these descriptions provide a general idea of how the graph will look. The specific shape of the graph may vary based on the actual function f(x) chosen to satisfy the given conditions.