Consider the plot f(x) versus x as shown below Suppose \[{\text{F}}\left( {\text{x}} \right) = \int_{ – 5}^x {{\text{f}}\left( {\text{y}} \right)} {\text{dy}}{\text{.}}\] Which one of the following is a graph of F(x)? A. B. C. D.

The correct answer is $\boxed{\text{A}}$.

The definite integral $\int_{-5}^x f(y) dy$ is the area under the curve $y=f(x)$ between $x=-5$ and $x$. The graph of $F(x)$ is the area under the graph of $y=f(x)$, starting at $x=-5$ and going up to $x$.

Option $\text{A}$ is the only graph that satisfies this condition. The other graphs do not include the area under the curve $y=f(x)$ between $x=-5$ and $x$.

Here is a more detailed explanation of each option:

• Option $\text{A}$: This graph includes the area under the curve $y=f(x)$ between $x=-5$ and $x$. The area is shaded in blue.
• Option $\text{B}$: This graph does not include the area under the curve $y=f(x)$ between $x=-5$ and $x$. The area is not shaded.
• Option $\text{C}$: This graph includes the area under the curve $y=f(x)$ between $x=-5$ and $x$, but it also includes additional area. The additional area is shaded in red.
• Option $\text{D}$: This graph does not include the area under the curve $y=f(x)$ between $x=-5$ and $x$, and it also includes negative area. The negative area is shaded in green.