A graph is concave up on an interval when its slope is increasing as you move left to right, so the curve looks like a “cup” or a U shape.

Intuitive idea (no formulas yet)

Think of driving along a road:

  • If you’re turning so that the road bends upward like a bowl, it’s concave up.
  • The tangent line (the local straight-line approximation) lies below the curve in a concave up region.
  • The function may be going up or down overall; concavity is about how the slope is changing, not just whether the function increases or decreases.

A classic example is y=x2y=x^2y=x2: its graph is always bending upward like a U, so it is concave up everywhere.

Formal calculus definition

For a differentiable function f(x)f(x)f(x):

  • The graph is concave up on an interval if the first derivative f′(x)f'(x)f′(x) is increasing on that interval.
  • Equivalently, if the second derivative f′′(x)f''(x)f′′(x) exists and
    • f′′(x)>0f''(x)>0f′′(x)>0 on an interval, then the graph is concave up there.

This gives the standard test used in calculus:

  • Find f′′(x)f''(x)f′′(x).
  • Find where f′′(x)>0f''(x)>0f′′(x)>0; those x-values are where the graph is concave up.

Visual and derivative tests side by side

Here’s a quick reference:

html

<table>
  <tr>
    <th>Perspective</th>
    <th>Concave Up Condition</th>
  </tr>
  <tr>
    <td>Visual shape</td>
    <td>Graph looks like a “cup”/U, opening upward.[web:1][web:9]</td>
  </tr>
  <tr>
    <td>Tangent lines</td>
    <td>Tangent lines lie below the graph on that interval.[web:7]</td>
  </tr>
  <tr>
    <td>Slopes</td>
    <td>Slopes of tangent lines increase as x increases.[web:3][web:5]</td>
  </tr>
  <tr>
    <td>First derivative</td>
    <td>\(f'(x)\) is increasing on the interval.[web:3][web:5]</td>
  </tr>
  <tr>
    <td>Second derivative</td>
    <td>\(f''(x) &gt; 0\) on the interval.[web:7][web:6]</td>
  </tr>
</table>

Example to lock it in

Take f(x)=x2f(x)=x^2f(x)=x2:

  • f′(x)=2xf'(x)=2xf′(x)=2x (this is increasing in x).
  • f′′(x)=2f''(x)=2f′′(x)=2, which is always positive.

So f(x)=x2f(x)=x^2f(x)=x2 is concave up for all real x, and its graph has that familiar U-shape. TL;DR:
A graph is concave up on an interval when it bends upward like a U, its slopes increase as you move right, and (in calculus terms) its second derivative is positive on that interval.

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