Definite Integrals Question 14

If $f'(x)\gt 0$ for all real numbers $x$ and $\int\limits_8^{10} f(t) dt = 0$ , which of the following could be a table of values for the function $f$ ?

\begin{array}{|c|c|} \hline x & f(x) \\ \hline \hline 8 & 0 \\ \hline 9 & 0 \\ \hline 10 & 0 \\\hline \end{array}

\begin{array}{|c|c|} \hline x & f(x) \\ \hline \hline 8 & 4 \\ \hline 9 & 6 \\ \hline 10 & -3 \\\hline \end{array}

\begin{array}{|c|c|} \hline x & f(x) \\ \hline \hline 8 & 2 \\ \hline 9 & 4 \\ \hline 10 & 6 \\\hline \end{array}

\begin{array}{|c|c|} \hline x & f(x) \\ \hline \hline 8 & -8 \\ \hline 9 & 1 \\ \hline 10 & 3 \\\hline \end{array}

Approach

The graph of $f$ is always increasing, which rules out the first two options.

We are given that the value of the integral is $0$ , which implies that the area under the curve must consist of positive and negative regions.

Only the last option correctly includes points below and above the $x$ -axis, indicating a possible area sum of $0$ .