Intermediate Value Theorem (Darboux's Property)
Say if the following equations have any solution using the Darboux property.
a) $x^2=1$
b) $e^x=\ln x+3$
c) $x^4+2x=0$
a) We define the function $f(x)=x^2$.
Taking the interval $[0,2]$ it is satisfied that $1$ belongs to the image interval $f([0,2])=[0,4]$, therefore a point $c$ exists such that $f (c) = 1$ and therefore a solution exists. (in our case $c=1$).
b) We define the function $f(x)=e^x-\ln x$.
Taking the interval $[1,2]$ it is satisfied that $3$ belongs to the image interval $f([1,2])=[2.7182\ldots,6.69\ldots]$ therefore a point $c$ exists such that $f (c) = 3$ and we can say that there exists some solution to our equation.
c) We define the function $f(x)=x^4+2x$ and repeat the process:
Taking the interval $[-1,1]$ it is satisfied that $0$ belongs to the image interval $f([-1,1])=[-1,3]$. Therefore in the interval $[-1,1]$ there exists a point that solves our equation.
a) It has at least one solution in the interval $[0,2]$.
b) It has at least one solution in the interval $[1,2]$.
c) It has at least one solution in the interval $[-1,1]$.