1. **Problem Statement:** We have a function $g$ defined on the interval $-11 \leq x \leq 7$ by $$g(x) = 8 + \int_{-11}^x f(t) \, dt,$$ where $f$ is a given function with areas under the curve labeled $A, B, C, D, E$ corresponding to intervals $[-11,-9], [-9,-5], [-5,-1], [1,5], [5,7]$ respectively. The areas are given as: - $A = 2$ - $B = 14$ - $C = 10$ - $D = 27$ - $E = 4$ We are asked to find the absolute minimum and maximum values of $g$ on $[-11,7]$ and justify the answer. 2. **Formula and Important Rules:** The function $g(x)$ is defined as an integral of $f$ from $-11$ to $x$ plus 8. By the Fundamental Theorem of Calculus, the derivative is: $$g'(x) = f(x).$$ The value of $g(x)$ can be found by adding the signed area under $f$ from $-11$ to $x$ to the initial value 8. 3. **Step-by-step Solution:** - At $x = -11$, $$g(-11) = 8 + \int_{-11}^{-11} f(t) dt = 8 + 0 = 8.$$ - Moving from $-11$ to $-9$, the area $A = 2$ is added: $$g(-9) = 8 + 2 = 10.$$ - From $-9$ to $-5$, area $B = 14$ is added: $$g(-5) = 10 + 14 = 24.$$ - From $-5$ to $-1$, area $C = 10$ is added: $$g(-1) = 24 + 10 = 34.$$ - Note the gap from $-1$ to $1$ is not mentioned, so assume no area or zero contribution. - From $1$ to $5$, area $D = 27$ is added: $$g(5) = 34 + 27 = 61.$$ - From $5$ to $7$, area $E = 4$ is added: $$g(7) = 61 + 4 = 65.$$ 4. **Finding Absolute Minimum and Maximum:** - The values of $g$ at the key points are: - $g(-11) = 8$ - $g(-9) = 10$ - $g(-5) = 24$ - $g(-1) = 34$ - $g(5) = 61$ - $g(7) = 65$ - Since $g$ is continuous and increasing by positive areas, the minimum value is at the start $x = -11$ with $g(-11) = 8$. - The maximum value is at the end $x = 7$ with $g(7) = 65$. 5. **Justification:** - Because $g'(x) = f(x)$ and the areas are positive, $g$ is increasing on the intervals where $f$ is positive. - No negative areas are given, so $g$ does not decrease. - Therefore, the absolute minimum is at the left endpoint and the absolute maximum at the right endpoint. **Final answer:** - Absolute minimum value of $g$ is $8$ at $x = -11$. - Absolute maximum value of $g$ is $65$ at $x = 7$.