Minimum of a Parabola

Problem

If , what is the minimum value of for real numbers ?

Step 1: Recognize the function type

This is a quadratic function:

It opens upward (since ), so it has a minimum at the vertex.

Step 2: Use the vertex formula

The vertex of a parabola is at:

Step 3: Plug in the vertex to find the minimum

Final Answer:

The minimum value of is:

Visual Insight

Vertex