Calculate Negative Area: Finding Area Below f(x) = x² + 16

Find the negative area of the function

$f(x)=x^2+16$

To solve this problem, we'll follow the steps outlined in our analysis.

• Step 1: Analyze the function's form $f(x) = x^2 + 16$. Here, $a=1, b=0,$ and $c=16$.

• Step 2: Find the vertex to see if the function ever takes negative values. The vertex $x$ is calculated by $x = -\frac{b}{2a} = 0$.

• Step 3: Evaluate $f(x)$ at this vertex: $f(0) = 0^2 + 16 = 16$.

• Step 4: Determine when $f(x) < 0$. Since $f(x) = x^2 + 16 \geq 16$ for all real numbers $x$, the function is always positive.

• Step 5: Compare the finding against multiple-choice options. The choice indicating that $f(x)$ is always positive is the correct one: "Always positive".

The conclusion, therefore, is as follows: the function $f(x) = x^2 + 16$ is always positive, and there is no negative area under the graph relative to the x-axis.