What does the second derivative of a quadratic function actually mean?

Question

The second derivative of $y=ax^2 +bx +x$ is $\frac{d^2y}{dx^2}=2a$

But what does $2a$ mean in terms of the graph of this function?

Take the function $f(x)=\frac{x^2}{2}$

It has a $2a$ value of $1$.

I understand that the second derivative expresses the concavity of a graph, but I can't see how a concavity of $1$ makes sense for this graph.

There have been posts similar to this topic but I have not seen a satisfactory answer.

Can anyone explain what it means to say that $f(x)=\frac{x^2}{2}$ has a concavity of $1$ everywhere along the graph?

Answer 1

Well, it means that the slope is increasing at a constant rate of 1 for positive $x$, similarly, it's decreasing at a constant rate of 1 for negative $x$.

Equivalently, the graph increases at an increasing rate (for positive $x$).

Compare this with the line $y=x$ which has $y''=0$. This means that the slope doesn't increase, which is what you see. The slope is constant.

Answer 2

We have that

• if $\frac{d^2y}{dx^2}=2a>0 \implies$ $f(x)$ is concave up (that is convex)

• if $\frac{d^2y}{dx^2}=2a<0 \implies$ $f(x)$ is concave down (that is concave)