Concavity of an integral function

[gohou@econ.umd.edu]

Hi guys,
I am new in this forum and really hope that somebody can help me.
I would like to show that the following function in concave of convex so I need to calculate the second derivative of Q(t) :

Q(t)=Integral[upper=T; Lower=0]{tF(t)q(p-st)}dt

 

[HallsofIvy]

I assume you mean Q(T). t is just the "dummy" variable of integration.

If $$Q(t)= \int_0^t (\tauF(\tau)q(p-s\tau))d\tau$$
then the best I can do is
$$\frac{dQ}{dt}= tF(t)q(p-st)$$
(by the fundamental theorem of calculus) without knowing what F is. (And is q a constant or a function?)

 

[tacman]

Concavity of an integral function refers to the shape of the graph of the function. A concave function has a graph that curves downward, while a convex function has a graph that curves upward. In order to determine the concavity of an integral function, we need to calculate its second derivative.

In this case, the function Q(t) can be rewritten as:

Q(t) = ∫[0,T] tF(t)q(p-st)dt

To calculate the second derivative, we can use the chain rule and the fundamental theorem of calculus. The first derivative of Q(t) is:

Q'(t) = tF(t)q(p-st) + ∫[0,T] F(t)q'(p-st)(-s)dt

Using the chain rule, we can calculate the second derivative:

Q''(t) = F(t)q(p-st) + tF'(t)q(p-st)(-s) + F(t)q'(p-st)(-s) + ∫[0,T] F(t)q''(p-st)(-s)^2dt

Simplifying the second derivative, we get:

Q''(t) = F(t)[q(p-st) - sq'(p-st)] + ∫[0,T] F(t)q''(p-st)(-s)^2dt

In order to determine the concavity of Q(t), we need to look at the sign of the second derivative. If Q''(t) is positive, then the function is convex. If Q''(t) is negative, then the function is concave. If Q''(t) is equal to zero, then the function is neither concave nor convex.

In this case, we can see that Q''(t) contains both positive and negative terms, which means that the concavity of Q(t) will depend on the values of F(t), q(p-st), and q''(p-st). Without knowing the specific values of these functions, we cannot determine the concavity of Q(t). However, we can use this second derivative to analyze the concavity at specific points or intervals.

I hope this helps in understanding the concavity of an integral function. If you have any further questions, please feel free to ask. Welcome to the forum and good luck with your calculations!