Convex Functions and Inequalities A convex function is a function whose graph is always concave upwards. This means that the graph forms a "bowl-like" s...
Convex Functions and Inequalities
A convex function is a function whose graph is always concave upwards. This means that the graph forms a "bowl-like" shape when viewed from above.
Examples of convex functions:
S^2 (the parabola)
x^2 + y^2
x^3 - 3x + 2
x^4 - 4x^2 + 4
Concave functions have the opposite property, with the graph always concave downwards.
Examples of concave functions:
x^2 - 1
x^4 - 2x^2 + 1
(x+1)^2
x^3 - 3x
Properties of convex and concave functions:
The first derivative of a convex function is always positive.
The second derivative of a concave function is always negative.
The graph of a convex function is always concave upwards.
The graph of a concave function is always concave downwards.
Important inequalities related to convex and concave functions:
f(x) ≤ f(y) for all x and y in the domain of f.
f(x) ≥ f(y) for all x and y in the domain of f.
f(x) > f(y) for x > y in the domain of f.
These inequalities can be used to determine the relative positions of points on the graph of a function