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Friday, 26 August 2011

Math GRE - #25

Let $k$ be the number of roots to the equation $f(x)=e^x+x-2$ in the interval $[0, 1]$, and let $n$ be the number of roots that are not in $[0,1]$. Which of the following is true?

  • $k=0$ and $n=1$
  • $k=1$ and $n=0$
  • $k=n=1$
  • $k>1$
  • $n>1$

Solution :

Choice 2 is the answer.

We can solve this problem by using the Intermediate Value Theorem.
By evaluating $f$ at 0 and 1, we see that: \[f(0)=e^0+0-2<0\] \[f(1)=e^1+1-2>0.\]
By the Intermediate Value Theorem, there exists a $c\in[0,1]$ such that $f(c)=0.$ Thus we know for sure that there is at least one root in [0,1].

Taking the derivative of $f$, we note that: \[f^\prime(x)=e^x+1>0\] for all $x$.
Thus $f$ is increasing everywhere and so it must only have one root. Since this root is in [0,1], $k=1.$ As we cannot have roots anywhere outside of [0,1], we also have $n=0$. This gives choice 2 as the answer.

Bonus question: The Intermediate Value Theorem may seem obvious, but it can also be used to prove many fun facts. One such fact is that if you draw a circle anywhere in the universe, there will be two points directly opposite to each other on the circle (i.e. 180 deg. apart) that have the exact same temperature. Can you prove this?

If you get stuck, here is some intuition as to why this interesting fact is true.


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