The shape of the functions

Question

1. How many of the following functions on R are increasing on their domain? $y = e^x$, $y = x^2$, $y = x^3$ (a) 0 (b) 1 (c) 3 (d) 2
2. How many of the following functions on R are concave up on their domain? $y = e^x$, $y = x^2$, $y = x^3$ (a) 3 (b) 1 (c) 0 (d) 2

So I have these two questions above from a past exam, the solution for both of them is D, but I was very certain that only $e^x$ fullfilled the first question, this is because $x^3$ and $x^2$ also decrease in their domain as well as they increase; for example at the point (-1,1) of $x^2$ the function is decreasing; for $x^3$ at the point (0,0) the function is flat, and even $e^x$ remains flat at some point on its domain before starting growing up exponentially...regarding question 28 I think that only 1, this is $x^2$ has a concave up shape in their domain....I am a bit confused here, can anyone shine some light on it?

Thanks :)

Answer 1

You have to be careful and precise with what you mean by "flat."

It's true that the derivative of $x^3$ is zero at $0$, but $x^3$ is still a (strictly) monotonic function: for any $a>b$, $a^3 > b^3$.

You can prove this directly: for $a>b$, $a-b > 0$ and $$a^3-b^3 = (a-b)(a^2+ab+b^2) \geq \begin{cases}(a-b)(a^2+b^2) > 0 & ab \geq 0\\(a-b)[(a+b)^2-ab]>0 & ab < 0.\end{cases}$$

Alternatively, you can use calculus: for $a>b$, $$f(b)-f(a) = \int_a^b f'(x)dx > 0,$$ if $f'$ is continuous and positive at all but countably many points. (Intuitively, there is always positive total area under the curve, even if the curve touches the $x$-axis at a few points.)