The equation $e^{x^3-x} - 2 = 0$ has solutions...

Question

The equation $$e^{x^3-x}-2$$

1. has no solution in $[0,1]$
2. has a unique solution in $[0,1]$
3. has two solutions in $[0,1]$
4. has four solutions in $[0,1]$

Correct answer is B but how do you approach a problem like this? Can you post a solution please? What does $[0,1]$ even mean?

Thank you

(thanks for the edit)

Answer 1

you have asked " What does $[0, 1]$ mean?"

It means all $x$ real numbers $x$ such that $ 0\le x\le 1$

In order to solve $$e^{x^3-x} =2 $$ We take logarithms to get $$ {x^3-x} = \ln2$$

We are looking for solutions in the closed interval $ [0,1].$

Note that for $ x\in [0, 1]$

$$ {x^3-x} \le 0$$ and $\ln(2) =0.6931...$ is positive.

Thus there is no solution between $0$ and $1$. The option $1$ is the correct answer.

Answer 2

$e^{x^3-x}=2=e^{ln 2}$

⇒ $x^3-x=ln 2$

$x ≈ 1.245$

Hence this equation has no solution in $[0, 1].$

Answer 3

A simple way to get the answer is to note that $x^{3}-x\leq 0$ and hence $e^{x^{3}-x} \leq 1 <2$.