Sketch the graph of a function f that satisfies all the following conditions.
(a) f(0)=0,f'=1
(b) f''(x)>0 for x<0, f''<0 for x>0
To sketch the graph of a function that satisfies all the given conditions, we need to use the information about the initial value, the derivative, and the concavity of the function.
(a) f(0) = 0, f' = 1:
Since f(0) = 0, the y-intercept of the graph will be at the point (0, 0). Additionally, since f' = 1, we know that the slope of the graph is always positive.
(b) f''(x) > 0 for x < 0, f'' < 0 for x > 0:
This condition tells us about the concavity of the graph. When f''(x) > 0 for x < 0, it means the graph is concave up (opening upwards) on the left side. When f''(x) < 0 for x > 0, it means the graph is concave down (opening downwards) on the right side.
Now, let's combine all this information to draw the graph:
1. Start by plotting the point (0, 0) as the y-intercept.
2. Since f' = 1, the graph will have a positive slope at every point. Thus, draw a line that starts at (0, 0) and goes upward.
For the concavity:
3. Draw a part of the graph to the left of 0, which is concave up. This means the graph should curve upwards in that region.
4. Draw a part of the graph to the right of 0, which is concave down. This means the graph should curve downwards in that region.
By following these steps, you will be able to sketch a graph that satisfies all the given conditions. Remember that the sketch is just an approximation, and you can refine it based on any additional information provided or any specific intervals mentioned in the problem.