On what interval is the curve y=0xt2t2+t+2dt concave downward?
The interval where the curve is concave downward
i) i) Fundamental Theorem of calculus. If is continuous on and
ii) Concavity test:
If on an interval, then is concave upward on that interval.
If on an interval, then is concave downward on that interval.
iii) Quotient rule:
If f and g are differentiable, then
The givencurve is
Differentiating both sides,
Then by fundamental theorem,
By using quotient rule,
Find such that .
These are the critical numbers of and they divide the domain into three intervals. When or we have Since when , we may divide the real line into intervals with these numbers as endpoints and construct the following chart.
Therefore, on interval the curve is concave downward
Therefore, the curve is concave up on the interval