Problem

Use the fact that \((x-5)^{2}\ge~0\), with equality at a specific input, to analyze \(f(x)=(x-5)^{2}+3\). State the extremum type, value and location, vertex, and range.

Big Picture

What this problem is really about

The squared term can contribute no less than zero, and it reaches zero only at the center input. We’ll shift that bound by the outside constant, use the equality case to identify the attained extremum and vertex, and include the boundary value when expressing the range.

Answer choices

Choose your path.

Preview only

The function has minimum 3 at x = 5 and vertex (5, 3), but range (3, ∞). The endpoint 3 is attained and must be included.

The function has minimum 1 at x = 5, vertex (5, 1), and range [1, ∞). Substituting x = 5 gives 3, not 1.

(x − 5)² ≥ 0, with equality at x = 5. The function has minimum 3 at x = 5, vertex (5, 3), and range [3, ∞).

The function has minimum 3 at x = −5, vertex (−5, 3), and range [3, ∞). The equality input is x = 5.

The choices are visible. Checking your work and seeing which answer is correct happen inside Gozunta.

Use the fact that \((x-5)^{2}\ge~0\), with equality at a specific input, to analyze \(f(x)=(x-5)^{2}+3\). State the extremum type, value and location, vertex, and range.

What this problem is really about

Turn the preview into practice.

Gozunta lets you study this problem—and more than 10,000 others.