Say if the following equations have a solution using Bolzano’s theorem:
a) $$x^2=1$$
b) $$e^x= 3+\ln x$$
c) $$x^4+2x=0$$
Development:
a) We define the function $$f(x)=x^2-1$$. We are going to look for two values $$a$$ and $$b$$ such that once we evaluate the function $$f (x)$$ we obtain values with opposite signs:
Taking
$$x=0 \Rightarrow f(0)=-1 < 0$$
$$x=2 \Rightarrow f(2)=5 > 0$$
so in the interval $$[0,2]$$ a point $$c$$ exists such that $$f (c) = 0$$ and therefore $$c$$ is a solution to our equation. (in this case $$c=1$$ and $$f (1) = 0$$).
b) We define the function $$f(x)=e^x-\ln x-3$$. Let's look for two values $$a$$ and $$b$$ such that once we evaluate the function $$f (x)$$ we obtain values with opposite signs:
Taking
$$x=1 \Rightarrow f(1)=e-0-3=-0.2817 < 0$$
$$x=2 \Rightarrow f(2)=3.69 > 0$$
So in the interval $$[1,2]$$ a point $$c$$ exists where $$f (c) = 0$$ and we know with certainty that some value that solves our equation exists.
c) We define the function $$f(x)=x^4+2x$$ and repeat the process:
Taking
$$x=-1 \Rightarrow f(-1)=(-1)^4+2\cdot(-1)=1-2=-1 < 0$$
$$x=1 \Rightarrow f(1)=1+2=3 > 0$$
so in the interval $$[-1,1]$$ there exists a point $$c$$ that is a solution to 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 $$[-1,1]$$.