when is a graph concave up second derivative

A graph of a function y=f(x)y=f(x)y=f(x) is concave up on an interval when its second derivative is positive on that interval, that is f′′(x)>0f''(x)>0f′′(x)>0.

Intuition: What “concave up” means

• Concave up looks like a cup or a bowl opening upward; tangent lines lie below the curve.

• As you move to the right, the slope of the graph is increasing (it may be going up faster, or going down more slowly).

A classic example is f(x)=x2f(x)=x^2f(x)=x2, which has f′′(x)=2>0f''(x)=2>0f′′(x)=2>0 everywhere, so its graph is concave up for all real xxx.

Second derivative rule (the key test)

Let fff be twice differentiable on an interval.

• If f′′(x)>0f''(x)>0f′′(x)>0 for all xxx in an interval, then the graph of y=f(x)y=f(x)y=f(x) is concave up on that interval.

• If f′′(x)<0f''(x)<0f′′(x)<0 on an interval, the graph is concave down there.

• Points where f′′(x)f''(x)f′′(x) changes sign (from + to − or − to +) are candidates for inflection points , where concavity changes.

So, “when is a graph concave up?”
Exactly when the second derivative is positive on that region: f′′(x)>0f''(x)>0f′′(x)>0.

Information gathered from public forums or data available on the internet and portrayed here.