Theorems regarding increasing, decreasing,and constant functions

Question

So my teacher would like us to prove the constant value theorem using the increasing and decreasing value theorem. I'm unsure how to do this since it would seem logical in each case individually but not when they are applied together. As a function increases then the derivative is increasing or greater than 0 and vice versa for decreasing. But to use this to prove the constant value theorem I am at a loss. Please help me. 

Answer 1

your teacher needs a 'constant value theorem' to be proven?  let f(x) = k

We know the graph is all ordered pairs (real,k). This function is neither decreasing nor increasing.

f'(x) = 0; therefore the tangents are all horizontal; this confirms that f is neither decreasing nor increasing.

 

You wrote: As a function increases then the derivative is increasing or greater than 0 and vice versa for decreasing. Clarification: as x increases for an increasing function, y increases; this merely requires that y' > 0; such derivative could be constant, in which case it would not be proper to say that then the derivative is increasing