2. (2 points) let f and g be functions defined on the set of all real numbers.\na. is the statement true or…

2. (2 points) let f and g be functions defined on the set of all real numbers.\na. is the statement true or false?\nif f(x) > g(x) for all x, then f(x) > g(x) for all x.\nb. if the statement is true, explain how you know. if the statement is false, give an example.

2. (2 points) let f and g be functions defined on the set of all real numbers.\na. is the statement true or false?\nif f(x) > g(x) for all x, then f(x) > g(x) for all x.\nb. if the statement is true, explain how you know. if the statement is false, give an example.

Answer

Explanation:

Step1: Consider the difference function

Let (h(x)=f(x)-g(x)). Then (h^{\prime}(x)=f^{\prime}(x)-g^{\prime}(x)). Given (f^{\prime}(x)>g^{\prime}(x)) for all (x), so (h^{\prime}(x)>0) for all (x). This means (h(x)) is an increasing function.

Step2: Analyze the relationship between (h(x)) values

Just because (h(x)) is increasing does not mean (h(x)>0) for all (x). For example, if (h(x)=x - 2), (h^{\prime}(x)=1>0) (so (h(x)) is increasing), but when (x = 1), (h(1)=1 - 2=-1<0).

Answer:

a. False. b. As shown in the example above, even if the derivative of (h(x)=f(x)-g(x)) (i.e., (h^{\prime}(x)=f^{\prime}(x)-g^{\prime}(x))) is positive (so (h(x)) is increasing), (h(x)) (and thus (f(x)-g(x))) can be negative for some (x). For instance, let (f(x)=x) and (g(x)=2). Then (f^{\prime}(x) = 1), (g^{\prime}(x)=0), so (f^{\prime}(x)>g^{\prime}(x)) for all (x). But when (x = 1), (f(1)=1) and (g(1)=2), and (f(1)<g(1)).