Theorem of Bolzano

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.