True or false about monotonic functions

Question

So I have a homework question that asks " A function which is monotonic on an interval is either increasing or decreasing on the interval." true or false?

I answered True because wouldn't it be by definition either increasing or decreasing?

 

So then part b stems off an asks true or false about x^3 being monotonic.

For this I said true but I wasn't sure because of when it crosses the origin.

 

Then part C asks true or false if x^2 is monotonic and I said false because when it touches the origin isn't it at zero, neither increasing or decreasing?

 

I'm a little confused mostly because of the word monotonic. Please help me and correct me and explain why if you do.

Answer 1

A monotonically increasing function never decreases. Meaning it could have a positive or zero slope.

 

A monotonically decreasing function never increases. It could have a zero or negative slope.

 

Thereofere, a monotonically increasing function doesnt always increase. A monotonically decreasing function isn't always decreasing.

Answer 2

A function which is monotonic on an interval is either increasing or decreasing on the interval." true or false?

 

A monotonic function is either nonincreasing or nondecreasing on the interval.

For b>a  f(b)≥f(a)  if the function is nondecreasing  (never goes down, but can stay the same)

For b>a  f(b)≤f(a)  if the function is nonincreasing   (never goes up, but can stay the same)

 

A horizontal line that never goes up or down is indeed

monotonic but is both nonincreasing and nondecreasing.

 

Since the function can have a slope of zero as with y=5 then it is

possible for a monotonic function to be neither decreasing nor increasing

over a particular interval.  Thus, this is false.

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Is y = x³ monotonic?

 

This function always has a positive slope on the interval (-∞, ∞) so it

is a nondecreasing monotonic function

 

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Is  y = x² a monotonic function?

 

This function is monotonic on the interval [0,∞) where it always increases.

It is monotonic on the interval (-∞,0] where it always decreases....

But on the interval (-∞, ∞) it increases on the positive side of zero

and decreases on the negative side of zero so it is not monotonic.