Suppose f is a linear function with positive slope. Show...

Suppose $f$ is a linear function with positive slope. Show that the average rate of change of $f$ on any interval $[a, b]$ is positive, and then use this fact to show that $f$ is always increasing.

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Key Concepts

Linear Function

A linear function is one of the form f(x) = mx + b, where m and b are constants. Its graph is a straight line, and the behavior of the function is fully determined by the slope m and the y-intercept b. This concept is fundamental in understanding how changes in x relate to changes in f(x).

Positive Slope

A positive slope (m > 0) indicates that as the independent variable x increases, the function f(x) also increases. This property is essential in recognizing the directional behavior of linear functions, ensuring that the function is rising as it moves from left to right.

Average Rate of Change

The average rate of change of a function on an interval [a, b] is calculated as (f(b) - f(a))/(b - a). For a linear function, this value is constant and equal to the slope m. A positive slope guarantees that the average rate of change is positive over any interval.

Monotonic (Increasing) Function

An increasing function is one in which, for any two points a and b with a < b, it holds that f(a) < f(b). Showing that the average rate of change is positive on every interval confirms that the function is always increasing, which is a key characteristic of functions with a positive slope.

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