Question

Determine whether the statement is true or false. If f and g are positive increasing functions on an interval I, then fg is increasing on I. a. True b. False

Answer 1

Answer: false

Step-by-step explanation:

If f and g are increasing on I, this implies that f' > 0 on I and g' > 0 on I. That is both f' and g' have a positive slope. However,

Using product rule;

(fg)' = fd(g) + gd(f)

(fg)' = f * g' + f' * g

and although it is given that g' and f' are both positive we don't have any information about the sign of the values of the functions themselves(f and g). Therefore, if at least one of the functions has negative values there is the possibility that the derivative of the product will be negative. For example;

f = x, g = 5x on I = (-5, -2)

f' = 1 and g' =5 both greater than 0

f and g are both lines with positive slopes therefore they are increasing, but f * g = 5x^2 is decreasing on I.