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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 $t = 25$

enter image description here

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?

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  • $\begingroup$ By the way, can you derive the TR function from this TC equation if you know the price is $10? $\endgroup$ – Note Sep 16 '18 at 1:41
  • $\begingroup$ 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 $2t-14=0$. $\endgroup$ – Adam Bailey Sep 16 '18 at 12:49
  • $\begingroup$ @AdamBailey, Yes Adam, 1/3 t^3 $\endgroup$ – Note Sep 16 '18 at 23:19
  • $\begingroup$ 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? $\endgroup$ – Note Sep 16 '18 at 23:20
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"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.

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