Sometimes, we do not have concavity to help determine an explicit that shows continuity at a point. The figure shows the graph of
together with its tangent line at the inflection point .
What is for ?
What is for ?
By considering the graph near , find the largest so that whenever .
The graph of is concave down and decreasing when . We want to find the maximal so that, for a given , it follows that if then . This will, of course, be a function of .
What is the inverse function near ? Give an expression for .
Using the concavity of the graph of , determine which point is closer to 3, or , along the horizontal axis. (Do not evaluate them and consider only small .)
From your answers above, find an expression — in terms of — for the largest so that if then .
Find the largest such that if , then . Give your answer as a fraction involving and .
In the figure, we have the graph that is increasing and concave up. Next to are the points and for a small .
Which point is closer to along the horizontal axis, or ?
What is the distance between the nearest point and from your answer above? Give an expression that involves and absolute values.
Use your answers above to find such that whenever . Give your answer to 4 decimal places.
The figure shows the graph of around a point where and . Nearby are points and .
From the graph, looking at the -axis, which of the points and is nearest to ? Let this number be .
What is in terms of ?
The claim is that if and , then provided that is true for all small . What is the largest such ?
The condition was to ensure truth to the figure. Does the same work if ?
It would appear that all your arguments depended upon the fact that the graph of the function was concave up. Let . Find so that whenever . Assume that is positive.
Find the largest such that if , then . Give your answer as a fraction.
Find the largest such that if , then . Give your answer as a fraction involving .