Which one of the following statements is true?


If f is continuous on (a, b), then f attains an absolute maximum value and f attains an absolute minimum inside the interval (a, b).

If f '(x) > 0 on the interval (a, b), then f is decreasing on the interval (a,b)

If f and g are increasing on the interval (a, b), then f + g is increasing on (a, b).

None of these are true.

To determine which statement is true, we will examine each statement one by one:

Statement 1: If f is continuous on (a, b), then f attains an absolute maximum value and f attains an absolute minimum inside the interval (a, b).

To determine the truth value of this statement, we need to understand the concept of continuity. A function f is continuous on an interval (a, b) if it is defined at every point in the interval and there are no "jumps" or "holes" in the graph.

The statement correctly states that if a function is continuous on (a, b), then it attains an absolute maximum value and an absolute minimum value within that interval. This is known as the Extreme Value Theorem. Therefore, Statement 1 is true.

Statement 2: If f'(x) > 0 on the interval (a, b), then f is decreasing on the interval (a, b).

To determine the truth value of this statement, we need to understand the relationship between the derivative of a function and its behavior. If the derivative of a function is positive on an interval, it means the function is increasing on that interval, not decreasing. Therefore, Statement 2 is false.

Statement 3: If f and g are increasing on the interval (a, b), then f + g is increasing on (a, b).

To determine the truth value of this statement, we need to understand how the sum of two functions behaves based on the behavior of each individual function. If both f and g are increasing on (a, b), it means that as x increases, both functions increase.

When adding two functions, their sum will also increase if at any given point the sum of the individual function values is positive. Therefore, Statement 3 is true.

Given the analysis, Statement 1 is true, Statement 2 is false, and Statement 3 is true. Therefore, the correct statement among the given options is: "If f is continuous on (a, b), then f attains an absolute maximum value and f attains an absolute minimum inside the interval (a, b)."

The correct statement is: If f is continuous on (a, b), then f attains an absolute maximum value and f attains an absolute minimum inside the interval (a, b).