The average value of a function \( f \) over the interval \( [-1,2] \) is -4 , and the average value of \( f \) over the interval \( [2,7] \) is 8 . What is the average value of \( f \) over the interval \( [-1,7] \) ? (A) 2 (C) \( \frac{1}{2} \) (D) 14

To find the average value of \( f \) over the interval \( [-1, 7] \), we first need to use the formula for the average value of a function over an interval \([a, b]\): \[ \text{Average value} = \frac{1}{b-a} \int_a^b f(x) \, dx \] We know the average value of \( f \) from \([-1, 2]\) is -4, which gives us: \[ -4 = \frac{1}{2 - (-1)} \int_{-1}^{2} f(x) \, dx = \frac{1}{3} \int_{-1}^{2} f(x) \, dx \] Multiplying by 3, we find: \[ \int_{-1}^{2} f(x) \, dx = -12 \] Now, for the interval \([2, 7]\), the average value is 8: \[ 8 = \frac{1}{7 - 2} \int_{2}^{7} f(x) \, dx = \frac{1}{5} \int_{2}^{7} f(x) \, dx \] Multiplying by 5, we find: \[ \int_{2}^{7} f(x) \, dx = 40 \] Now, we can add these integrals to find the total integral over \([-1, 7]\): \[ \int_{-1}^{7} f(x) \, dx = \int_{-1}^{2} f(x) \, dx + \int_{2}^{7} f(x) \, dx = -12 + 40 = 28 \] The length of the interval \([-1, 7]\) is: \[ 7 - (-1) = 8 \] Now we calculate the average value over the interval \([-1, 7]\): \[ \text{Average value} = \frac{1}{8} \int_{-1}^{7} f(x) \, dx = \frac{28}{8} = 3.5 \] Thus, the average value of \( f \) over the interval \([-1, 7]\) is not among the options provided. However, let's see which is the closest choice. None fits 3.5 exactly. If you want to make further calculations or rounds, revise based on context, or check provided options.