# Is this Cost function concave or convex?

Given the following cost function, where t is the quantity of some product.

$$C(t) = 1/3t^3 - 7t^2 +11t + 50$$

here is a graph between $t= 0$ and $t = 25$

We are asked if this function is convex or concave?

Technically it is neither, but I am asking, in the context of economics, do we have to define intervals and say, between $0$ and $4.11$, it is concave, and for $t>4.12$ we have a convex function?

• By the way, can you derive the TR function from this TC equation if you know the price is $10? – Note Sep 16 '18 at 1:41 • Do you mean$(1/3)t^3$? If so the second derivative is$2t-14$and the point of inflexion (change from concave to convex) is at$t$such that$2t-14=0$. – Adam Bailey Sep 16 '18 at 12:49 • @AdamBailey, Yes Adam, 1/3 t^3 – Note Sep 16 '18 at 23:19 • But I am starting to think this cost function does not even make sense. I think it should be$111t$instead of$11t$in the third term. One of the reasons this doesn't make sense: How come the TC drops below the Fixed cost? – Note Sep 16 '18 at 23:20 ## 1 Answer "In the context of economics", it is neither - because convexity in economics means the same it does in math. If you redefine its domain, such that it's only computed for any$t > 4.12\$, this function's image is clearly convex: no line between two vectors inside it isn't entirely contained in the set.

P.S: Although convexity has the same definition in both fields, it has an important economic interpretation and is a very important hypothesis in many classical general equilibrium models.