Prove that there exists a real number $x$ such that $x^{177} + \frac{165}{1+x^8+\sin^2(x)} = 125$ using Intermediate Value Theorem.
Uhhh I have no idea where to even start with this. Anything to give me an idea of what to do here would be great.
Asked
Active
Viewed 43 times
0
-
I suppose that you have been asked to show that there is at least one real number $x$ such that .... Because, there are three. – Claude Leibovici Apr 22 '21 at 07:25
1 Answers
5
Define $$f(x)=x^{177} + \frac{165}{1+x^8+\sin^2(x)}- 125.$$ Clearly $f(x)$ is continuous in $[-2,2]$. Since $f(-2)<0, f(2)>0$, there is $c\in(-2,2)$ such that $f(c)=0$.
xpaul
- 44,000
-
1$f(0) = 0^{177} + \frac{165}{1 + 0^8 + \sin^2(0)} - 125 = 0 + \frac{165}{1} - 125 = 40$, so $f(0) > 0$. This was probably a typo; it does hold that $f(1) < 0$, because $f(1) = 1 + \frac{165}{1 + 1 + \sin^2(1)} - 125 \leq 1 + \frac{165}{2} - 125 = -41.5$. That $\leq$ followed since $\sin^2(1) > 0$. Therefore, $f(1) < 0$, and $f(2) > 0$, so there is a $c \in (1, 2)$ such that $f(c) = 0$ by the IVT – Nicholas Priebe Apr 21 '21 at 21:40