What on Earth is going on?

Calculus Level 2

Given a polynomial $f: \mathbb{R}^+ \rightarrow \mathbb{R}^+$ that satisfies $f(0) = 1, f(1) = e, f(e) =\pi$ .

Let $\displaystyle I = \int_1^e \frac{ f'(x) } { f(x) } \, dx$ . Find the value of $\text{sgn}(I)$ .

Notation : $\text{sgn}(x)$ denote the signum function, where $\text{sgn}(x) = \begin{cases} \dfrac{x}{|x|} , x \ne 0 \\ 0 , x = 0 \end{cases}$ .

1 -1 0

1 solution

Otto Bretscher
Dec 1, 2015

An antiderivative of $\frac{f'(x)}{f(x)}$ is $\ln(f(x))$ . By the fundamental theorem, $I=\ln(f(e))-\ln(f(1))=\ln(\pi)-1>0$ with $sgn(I)=\boxed{1}$

There are only three possible cases in signum function.. i.e $sgn(x < 0)=-1 , sgn(x= 0) = 0 , sgn(x > 0) = 1$ .
And we, have 3 attempts to solve this question. It means there is 100 % probability of solving this question without actual solving.

Akhil Bansal - 5 years, 6 months ago

.... as long as people know that there are only three possibilities ;)

Otto Bretscher - 5 years, 6 months ago

Fixed by making it a multiple choice question :)

Calvin Lin Staff - 5 years, 6 months ago

Are we making any assumptions about the function? E.g. Must the function be continuous / differentiable?

Calvin Lin Staff - 5 years, 6 months ago

When I assign problems like this in my classes, I usually require $f$ to be a polynomial to cover all the domain and differentiability issues without sounding too technical. I would start with "Given a positive-values polynomial $f$ satisfying..."

Otto Bretscher - 5 years, 6 months ago

What on Earth is going on?

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