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Average value of a function practice problems

The average value of f on [a, b], and where f actually takes that value. 10 problems with worked solutions; in 10 of them every equation is proved by a computer algebra system.

Find the average value of \( \displaystyle f(x) = \frac{4}{x^{2}} \) on \( \displaystyle [1, 3] \), and every \( \displaystyle c \) in the interval with \( \displaystyle f(c) = f_{\text{ave}} \).

Problem 4.292medium✓ Every equation proved

Find the average value of \( \displaystyle f(x) = 2 x^{2} - 2 x + 3 \) on \( \displaystyle [-1, 2] \), and every \( \displaystyle c \) in the interval with \( \displaystyle f(c) = f_{\text{ave}} \).

Problem 4.293medium✓ Every equation proved

Find the average value of \( \displaystyle f(x) = 3 x^{2} + 2 x + 3 \) on \( \displaystyle [-1, 3] \), and every \( \displaystyle c \) in the interval with \( \displaystyle f(c) = f_{\text{ave}} \).

Problem 4.294medium✓ Every equation proved

Find the average value of \( \displaystyle f(x) = - x^{2} - 3 x - 1 \) on \( \displaystyle [-2, 1] \), and every \( \displaystyle c \) in the interval with \( \displaystyle f(c) = f_{\text{ave}} \).

Problem 4.295medium✓ Every equation proved

Find the average value of \( \displaystyle f(x) = x^{2} \) on \( \displaystyle [0, 2] \), and every \( \displaystyle c \) in the interval with \( \displaystyle f(c) = f_{\text{ave}} \).

Problem 4.296medium✓ Every equation proved

Find the average value of \( \displaystyle f(x) = - 3 x^{2} - x \) on \( \displaystyle [-2, 2] \), and every \( \displaystyle c \) in the interval with \( \displaystyle f(c) = f_{\text{ave}} \).

Problem 4.297medium✓ Every equation proved

Find the average value of \( \displaystyle f(x) = - 2 x^{2} - 2 x - 3 \) on \( \displaystyle [0, 1] \), and every \( \displaystyle c \) in the interval with \( \displaystyle f(c) = f_{\text{ave}} \).

Problem 4.298medium✓ Every equation proved

Find the average value of \( \displaystyle f(x) = \frac{1}{x^{2}} \) on \( \displaystyle [1, 3] \), and every \( \displaystyle c \) in the interval with \( \displaystyle f(c) = f_{\text{ave}} \).

Problem 4.299medium✓ Every equation proved

Find the average value of \( \displaystyle f(x) = x^{2} + 2 x - 1 \) on \( \displaystyle [0, 3] \), and every \( \displaystyle c \) in the interval with \( \displaystyle f(c) = f_{\text{ave}} \).

Problem 4.300medium✓ Every equation proved

Find the average value of \( \displaystyle f(x) = 2 \sqrt{x} \) on \( \displaystyle [0, 1] \), and every \( \displaystyle c \) in the interval with \( \displaystyle f(c) = f_{\text{ave}} \).

Problem 4.301medium✓ Every equation proved