sketch graph of a function f that is differentiable and that satisfies the following conditions:(1)f'(x)>0, when x<-5
(2)f'(x)<0, when -5<x<1 and when x>1
(3)f'(-5) =0and f'(1)=0
(4)f(-5)=6and f(1)=2.
To sketch the graph of the function f that satisfies the given conditions, you can follow these steps:
1. Determine the intervals where f'(x) is positive and negative:
- From condition (1), f'(x) is positive when x < -5.
- From condition (2), f'(x) is negative when -5 < x < 1 and when x > 1.
2. Identify the critical points of f'(x):
- From condition (3), f'(-5) = 0 and f'(1) = 0. These are the critical points where the function changes from increasing to decreasing or vice versa.
3. Determine the behavior of f(x) based on the sign of f'(x):
- Before -5, f'(x) is positive, indicating that f(x) is increasing.
- Between -5 and 1, f'(x) is negative, indicating that f(x) is decreasing.
- After 1, f'(x) is negative again, indicating that f(x) is decreasing.
4. Determine the y-values of f(x) at the critical points:
- From condition (4), f(-5) = 6 and f(1) = 2.
Based on these steps, you can sketch the graph of f(x) as follows:
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-5 1
In this sketch, the graph starts at (x = -5, y = 6) and goes downward to a minimum point at (x = 1, y = 2). The graph is increasing before x = -5 and decreasing after x = 1. The critical points, where the graph changes from increasing to decreasing, are marked with dots.