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Convex Function Verifier

Verify convexity and concavity using the second derivative test

Select Function Type

Quadratic ax^2+bx+c Cubic ax^3+bx^2 e^(ax) ln(x)

f(x)=x^2+x+

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.

Convex (f>=0)

x^2 (a>0): always convex. e^x: always convex (f=e^x>0). -ln(x): convex (f=1/x^2>0).

Concave (f<=0)

-x^2: always concave. ln(x): concave (f=-1/x^2<0). sqrt(x): concave (f<0 for x>0).

Neither

x^3: f=6x changes sign at 0. Sin(x): f=-sin(x) oscillates. Neither globally convex nor concave.

Optimization

Convex functions: any local minimum is global. Gradient descent converges to global optimum. Critical in machine learning.

Teaching Example: f(x)=x^2. f=2>0 always. f>=0 everywhere -> CONVEX (holds water). f(x)=-x^2: f=-2<0 always -> CONCAVE (sheds water). f(x)=x^3: f=6x changes sign at x=0 -> NEITHER.

Applications

Optimization Machine Learning Economics Engineering Statistics

Frequently Asked Questions

What is convex function?▼

f(x)>=0 always. Curves upward. Chord above graph. One global minimum. x^2, e^x are convex.

Convex vs concave?▼

Convex: f>=0 (cup). Concave: f<=0 (cap). x^2 convex, -x^2 concave. ln(x) concave.

Why convex matters?▼

Convex optimization guarantees global optimum. No local minima trap. Used in ML, economics, engineering.

Is x^3 convex?▼

No. f=6x changes sign at 0. x<0: concave (f<0). x>0: convex (f>0). Neither globally.

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