Let \[ f(x)=\left\{\begin{array}{ll} 0 & \text { if } x<-5 \\ 3 & \text { if }-5 \leq x<0 \\ -2 & \text { if } 0 \leq x<3 \\ 0 & \text { if } x \geq 3 \end{array}\right. \] and \[ g(x)=\int_{-5}^{x} f(t) d t \] Determine the value of each of the following: (a) $g(-7)=$ (b) $g(-4)=$ (c) $g(1)=$ (d) $g(4)=$ (e) The absolute maximum of $g(x)$ occurs when $x=$ and is the value It may be helpful to make a graph of $f(x)$ when answering these questions.

Step 1 ：The function \(g(x)\) is the integral of \(f(x)\) from \(-5\) to \(x\). This means that \(g(x)\) is the area under the curve of \(f(x)\) from \(-5\) to \(x\).

Step 2 ：To find the value of \(g(x)\) at a specific point, we need to calculate the area under the curve of \(f(x)\) from \(-5\) to that point.

Step 3 ：For \(g(-7)\), since \(-7 < -5\), the integral from \(-5\) to \(-7\) does not exist, so \(g(-7) = 0\).

Step 4 ：For \(g(-4)\), we need to calculate the area under the curve from \(-5\) to \(-4\). Since \(f(x) = 3\) for \(-5 \leq x < 0\), the area is a rectangle with height 3 and width 1, so \(g(-4) = 3 * 1 = 3\).

Step 5 ：For \(g(1)\), we need to calculate the area under the curve from \(-5\) to \(1\). This is the sum of the area from \(-5\) to \(0\) and the area from \(0\) to \(1\). The area from \(-5\) to \(0\) is a rectangle with height 3 and width 5, so it is \(3 * 5 = 15\). The area from \(0\) to \(1\) is a rectangle with height \(-2\) and width 1, so it is \(-2 * 1 = -2\). Therefore, \(g(1) = 15 - 2 = 13\).

Step 6 ：For \(g(4)\), we need to calculate the area under the curve from \(-5\) to \(4\). This is the sum of the area from \(-5\) to \(0\), the area from \(0\) to \(3\), and the area from \(3\) to \(4\). The area from \(-5\) to \(0\) is \(3 * 5 = 15\), the area from \(0\) to \(3\) is \(-2 * 3 = -6\), and the area from \(3\) to \(4\) is \(0 * 1 = 0\). Therefore, \(g(4) = 15 - 6 + 0 = 9\).

Step 7 ：The absolute maximum of \(g(x)\) occurs when the area under the curve is maximized. This happens at \(x = 0\), where the area under the curve from \(-5\) to \(0\) is maximized and is the value \(15\).

Step 8 ：Final Answer: (a) \(g(-7) = \boxed{0}\) (b) \(g(-4) = \boxed{3}\) (c) \(g(1) = \boxed{13}\) (d) \(g(4) = \boxed{9}\) (e) The absolute maximum of \(g(x)\) occurs when \(x = \boxed{0}\) and is the value \(\boxed{15}\).