Show that affine functions are both concave and convex

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SUMMARY

Affine functions, defined as f(x) = cTx for some vector c, are both concave and convex. This is established by demonstrating that the second derivative of an affine function is zero, indicating no curvature. Consequently, affine functions satisfy the definitions of both concavity and convexity, as they lie on a straight line without any bends.

PREREQUISITES
  • Understanding of affine functions and their mathematical representation
  • Basic knowledge of concavity and convexity in mathematical functions
  • Familiarity with vector notation and operations
  • Concept of derivatives and their role in determining function curvature
NEXT STEPS
  • Study the properties of affine functions in linear algebra
  • Learn about the definitions and implications of concave and convex functions
  • Explore the relationship between derivatives and function curvature
  • Investigate applications of affine functions in optimization problems
USEFUL FOR

Students studying calculus, mathematics enthusiasts, and anyone interested in understanding the properties of functions in linear algebra.

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Homework Statement


In my textbook, the author briefly makes a statement that affine functions are both concave and convex, how is that true? and how can it be proven?


Homework Equations





The Attempt at a Solution

 
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Isn't an affine function a line?
Lines don't have any concavity.
 
This is exactly how the problem goes
let f(x) be a function in Rn.
prove that f(x) is both concave and convex if f(x) = cTx for some vector c

I thought that the function was a affine function, but i can't prove it
 

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