Say if the following equations have any solution using the Darboux property.
a) $$x^2=1$$
b) $$e^x=\ln x+3$$
c) $$x^4+2x=0$$
Development:
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.
Solution:
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 $$[-1,1]$$.