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If the derivative of a function is bigger than or equal to zero on an interval, it is increasing.

If the derivative of a function is bigger than zero on an interval, it is strictly increasing.

Professor,

For HW11 #5(a), should it be show that f is increasing on I, not strictly increasing? The derivative of f is zero at pi/2. Strictly increasing requires the derivative to be greater than zero for all points on the interval.

f(x) = x^3 is strictly increasing on (-5, 5) and f'(0) = 0.

If f'(x) > 0 on I, then f is strictly increasing on I.

If f is strictly increasing on I, then f' is greater than or equal to 0 on I.

The book is making a distinction between strictly increasing and increasing. According to these definitions, #4 is strictly increasing, but #5 is only increasing. Just to clarify,

for #4, f(x) = x + 2(root2), and

for #5, f(x) = x - pi + cos x.

On page 249, problem 26.8 says f is increasing on I iff the derivative is bigger than or equal to 0 for all x in I.

On page 245, Theorem 26.8 says if the derivative is bigger than 0 for all x in I, then f is strictly increasing.

The bottom line is: strictly increasing does not imply f'(x)>0. See my example below.

How does it not imply the derivative is bigger than zero if that's what's in the definition?