Hi, I have a function question may you help me?
Which one is increasing function for all x values?
(a) y = |x - 7|
(b) y = 2x^2 + 9
(c) 3x^3 -11
(d) 4x^4 + 2 how can i prove?
Thank you
You posted this in the pre-calculus forum.
Thus there is no way to prove this one way or the other.
You can simply draw the graphs and see the answer. But that is hardly a poof.
On the other hand, with calculus we can see which one has a non-negative derivative. That would prove it.
Assigning a value to x has no indication of whether a function is increasing or not.
Why don't you just graph the function? To rigorously prove it, we note that its derivative is 9x^2, which is always non-negative. Therefore the function in (C) is always increasing (except when x = 0, where the derivative is zero).
Technically the function $\displaystyle f(x)=3x^3-11$ is increasing everywhere.
The the statement that $\displaystyle f$ is an increasing function means that if $\displaystyle a<b$ then $\displaystyle f(a)<f(b)$.
That is clearly true in this case. When I made the remark reply #3 about proof, it was addressing the fact that this is a precalculus forum.
There is of course a perfectly good way of proving this.
Suppose that $\displaystyle a<b$ then $\displaystyle a^3<b^3$ then $\displaystyle a^3-11<b^3-11$. Proved.