Implicit differentiation, concavity and convexity

What is an implicit function?

Dependent variable not isolated to one side of the equation

F(x,y) = 0

Generalisation of implicit function

1. Set Z = f(X,Y) = 0

2. Then dZ = ∂z/∂X dX + ∂z/∂Y dY = 0

3. dY/dX=∂z/∂x/∂z/∂y = - fx/fy

When is a function concave?

Looks like a cave

d²Y/dX²<0

When is a function convex?

d²Y/dX²>0

What is the point of inflection? How is it found?

Where curve changes from concave to convex - found by setting second order derivative to 0.

Increase point of inflection (X,Y) by ±0.5 to determine concavity and convexity

What is a turning point?

Where slope changes direction indicating change in direction of curve