Question

$$ \begin{array}{l}{\text { True or False If } f \text { is differentiable and } f^{\prime}(c)>0 \text { for every } c \text { in }} \\ {(a, b), \text { then } f \text { is increasing on }(a, b) . \text { Justify your answer. }}\end{array} $$

Short answer

True. If the derivative of a function \( f \) is greater than zero for every point in an open interval \( (a, b) \), then the function \( f \) is increasing on that interval.

Step-by-step solution

Understanding what the problem is asking

The problem is asking whether a function that is always differentiable and where its derivative is greater than zero is always increasing.

Recall the definition of an increasing function

A function \( f \) is said to be increasing on an interval \( (a, b) \) if for any two numbers \( x \) and \( y \) in \( (a, b) \), if \( x < y \) then \( f(x) < f(y) \).

Recall the meaning of the derivative at a point

The derivative of a function \( f \) at a certain point \( c \), denoted as \( f'(c) \), measures the rate at which the function is changing at that point. If \( f'(c) > 0 \), it means the function is increasing at \( c \). If this is true for every point in the interval \( (a, b) \), it would mean that the function is increasing at every point in that interval.

Conclusion

Since \( f' \) is greater than zero for every point in \( (a, b) \), it means the function \( f \) is increasing at every point in \( (a, b) \). Therefore, according to the definition, \( f \) is an increasing function on \( (a, b) \).