5. If a function \( f \) is continuous on the closed interval \( [a, b] \), then \( y=\frac{1}{b-a} \int_{a}^{b} f(x) d x \) is the of \( f \) over \( [a, b] \). 6. True or False If a function \( f \) is continuous on a closed interval \( [a, b] \) and if \( m \) and \( M \) denote the absolute minimum value and the absolute maximum value, respectively, of \( f \) on \( [a, b] \), then \[ m \leq \int_{a}^{b} f(x) d x \leq M \]

The expression \( y=\frac{1}{b-a} \int_{a}^{b} f(x) d x \) represents the average value of the function \( f \) over the interval \( [a, b] \). This concept links beautifully to our daily lives, as averages help us make sense of data—be it your average spending in a month or the average score in a game. So, whether you're calculating your grades or mixing ingredients in a recipe, averaging plays a key role! Regarding the statement presented, it is indeed true! The integral of a continuous function over a closed interval will always yield a value that lies between the function's absolute minimum (m) and maximum (M) on that interval. This property is foundational in calculus and ensures that we can make robust predictions about the behavior of continuous functions, whether it's in determining areas under curves or analyzing trends in business and science. So, keep this in mind for ensuring your computations with limits hold true!