Derivatives

[b!tcoin]

Hello everyone, i want to prove this
-Theorem 4.10 goes like this, Let ?:[?,?]→ℝ f:[a,b]→R be continuous and differentiable on (a,b). If ?′(?)≥0 for all x∈(a,b), then f is monotonically increasing. Likewise, f is strictly increasing, monotonically decreasing, and strictly decreasing if ‘≥’ is replaced by ‘>’, ‘≤’, and ‘<’, resp.


- I tried proving functions like x^2, x^3, sin(x) but no one seemed to work, because i want to prove that "IT IS NOT TRUE THAT f'(x)>0 for all x.
Any help will be greatly appreciated

 

[Steven G]

Try cubert(x)

 

[b!tcoin]

the cube root of x?

 

[Steven G]

the cube root of x?

Yes

 

[Steven G]

Even sqrt(x)

You just need to draw a function that has the property that you want and then try to think of one that looks like what you drew. Can you think of any others?

 

[pka]

Hello everyone, i want to prove this View attachment 15210
-Theorem 4.10 goes like this, Let ?:[?,?]→ℝ f:[a,b]→R be continuous and differentiable on (a,b). If ?′(?)≥0 for all x∈(a,b), then f is monotonically increasing. Likewise, f is strictly increasing, monotonically decreasing, and strictly decreasing if ‘≥’ is replaced by ‘>’, ‘≤’, and ‘<’, resp.
- I tried proving functions like x^2, x^3, sin(x) but no one seemed to work, because i want to prove that "IT IS NOT TRUE THAT f'(x)>0 for all x.
Any help will be greatly appreciated

A counterexample is the way to "prove the negative". Do you know of a counterexample?

 

[b!tcoin]

how is it possible to take x square? if i take the interval [-1, 1] f is strictly decreasing in [-1, 0] and i want to prove its strictly
increasing

 

[b!tcoin]

nope, if i knew i wouldnt ask here lol

 

[pka]

how is it possible to take x square? if i take the interval [-1, 1] f is strictly decreasing in [-1, 0] and i want to prove its strictly increasing

Surely you know that if \(\displaystyle f(x)=x^2\) then the function \(\displaystyle f\) is decreasing on \(\displaystyle (-\infty,0)\) but increasing on \(\displaystyle (0,\infty)\).

 

[Dr.Peterson]

Hello everyone, i want to prove this View attachment 15210
-Theorem 4.10 goes like this, Let ?:[?,?]→ℝ f:[a,b]→R be continuous and differentiable on (a,b). If ?′(?)≥0 for all x∈(a,b), then f is monotonically increasing. Likewise, f is strictly increasing, monotonically decreasing, and strictly decreasing if ‘≥’ is replaced by ‘>’, ‘≤’, and ‘<’, resp.


- I tried proving functions like x^2, x^3, sin(x) but no one seemed to work, because i want to prove that "IT IS NOT TRUE THAT f'(x)>0 for all x.
Any help will be greatly appreciated

Are you saying that each of the three functions in your list is strictly increasing over some particular interval? And are you saying that for each of them (over such an interval), f'(x)>0? You've missed the counterexample you need ...

Think again about x^3. Tell us exactly what your thoughts are about that function. What interval [a, b] did you use? What did you find about its derivative?

 

[b!tcoin]

x^3 on [-1, 1]

 

[Dr.Peterson]

That's the function, and that's one possible interval. What do you observe about it, with regard to continuity, differentiability, whether it is strictly increasing, and whether f'(x) > 0? You said "no one seemed to work"; tell us what went wrong.