i need to sketch a graph that satisfies all of these conditions:
- it is defined for all #s except 0
- f(1) = 0 f(2)=2 f(3)=1
- f'(2)=0 f''(3)=0
- f''(x)<0 when x<0 and when 0<x<3
- f''(x)>0 when x>3
- lim x->0 = -inf
- lim x->inf = 0
- lim x->-inf = 2
sorry about the organization, hope you can help!
I cant sketch here. Locate the points first, then sketch at those points the slope lightly, then note the points of infliction, when the curvature changes. The outside limits next, sketch those, then fill in all in between.
To sketch the graph that satisfies all of the given conditions, we'll go through the process step by step. Here's what you can do:
1. Identify the points that are given:
- f(1) = 0: This means that the point (1, 0) lies on the graph.
- f(2) = 2: This means that the point (2, 2) lies on the graph.
- f(3) = 1: This means that the point (3, 1) lies on the graph.
2. Determine the derivative points:
- f'(2) = 0: This implies that the slope of the graph is zero at x = 2.
3. Find the points of inflection:
- f''(3) = 0: This indicates that the graph changes its curvature at x = 3. However, the given conditions don't specify the exact nature of the inflection points, so for simplicity, we won't go into further detail about them in this explanation.
4. Determine the intervals of increase and decrease:
- f''(x) < 0 when x < 0 and 0 < x < 3: This implies that the graph is concave down in the interval (0, 3).
- f''(x) > 0 when x > 3: This indicates that the graph is concave up for x > 3.
5. Determine the vertical asymptotes:
- lim x -> 0 = -inf: This means that the graph has a vertical asymptote at x = 0.
- lim x -> inf = 0: This indicates that the graph has a horizontal asymptote at y = 0.
- lim x -> -inf = 2: This implies that the graph has a horizontal asymptote at y = 2.
Now let's proceed with sketching the graph:
- Start by plotting the points (1, 0), (2, 2), and (3, 1).
- At the points (1, 0) and (3, 1), lightly sketch the slopes based on the given concavity information. For example, the slope at x = 1 will be decreasing, and the slope at x = 3 will be increasing.
- Note the presence of vertical asymptote at x = 0, so the graph cannot cross this vertical line.
- Similarly, note the horizontal asymptotes at y = 0 and y = 2.
- Based on the given information about increase and decrease intervals, the graph will be concave down between x = 0 and x = 3, and concave up for x > 3.
- Connect the plotted points smoothly following the concavity and asymptotes.
- Make sure the graph approaches the asymptotes as x tends to infinity and minus infinity.
Remember, this is a general explanation, and the specific shape and behavior of the graph can vary depending on the details of the concavity and inflection points. Sketch lightly at first, and refine as needed. It's best to use graphing software or a graphing calculator to visualize the graph accurately.