Verify convexity and concavity using the second derivative test
Select Function Type
f(x)=x^2+x+
f(x)=x^3+x^2
f(x)=e^(x)
f(x)=ln(x) (domain: x>0)
Result
Derivation
Convexity Test
f(x) >= 0 for all x: convex (curving up)
f(x) <= 0 for all x: concave (curving down)
f changes sign: neither (mixed)
Linear functions: both convex and concave
A function is convex if the second derivative is non-negative everywhere. It is concave if the second derivative is non-positive. The second derivative measures curvature: positive f means the graph bends upward like a cup, the defining property of convex functions.
⚠For twice-differentiable functions: f>=0 everywhere means convex. f<=0 everywhere means concave. If f changes sign, the function is neither globally convex nor concave.
What Is Convexity?
Convexity describes the curvature of a function. A convex function bends upward (like a cup) and any chord lies above the graph. This property ensures a unique global minimum, making convex functions extremely important in optimization theory and machine learning.
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