Problem 97 - Essential Calculus

Question

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. The sum of two increasing functions is increasing.

Step-by-Step Solution

Answer

The statement is true. The sum of two increasing functions is also increasing.

Step 1: Understand the Definition

An increasing function means a function where its values increase as the input, \(x\), increases. In other words, if \(x_1 < x_2\) then \(f(x_1) \leq f(x_2)\) for an increasing function \(f.\)

Step 2: Consider the Sum of Two Increasing Functions

Assume \(f\) and \(g\) are two increasing functions. If we take two points \(x_1 < x_2\), since \(f\) and \(g\) are increasing functions, \(f(x_1) \leq f(x_2)\) and \(g(x_1) \leq g(x_2)\) should be true. When we add these inequalities, we get \(f(x_1) + g(x_1) \leq f(x_2) + g(x_2)\). This means the function \(h(x) = f(x) + g(x)\) also satisfies the definition of increasing function.

Step 3: Conclusion

Since the sum of the two increasing functions satisfies the definition of an increasing function, we can conclude that the statement 'The sum of two increasing functions is increasing' is true.