Q:

Applying Intermediate Value Theorem to a function

Question: $f(x$) = $x^5-15x^4-10x^2+20$
Prove that the equation $f(x) = 0$ has a real solution with $x \geq 1$.
Idea: Started off by coming up with a boundary $[1,x]$ for some real number $x$. $f(1)=-4$ so is less than zero, then attempted to find an $x$ that gave me a positive result, very quickly came to the conclusion that any number above 1 when inserted into a function gave me a negative number. Pretty sure I could be missing something really simple.
Any help or hints would be appreciated.

A:

$f(20)$ is positive. So, check again.
Also, for future purposes, to solve such questions, you can easily check the behaviour of a polynomial at large $x$ by just considering the sign of the highest power variable. If it is positive, then $f(x)\to\infty$, and if it is negative, then $f(x)\to-\infty$.

