Problems from Theorem of Bolzano

Say if the following equations have a solution using Bolzano’s theorem:

a) $x^{2} = 1$

b) $e^{x} = 3 + \ln x$

c) $x^{4} + 2 x = 0$

See development and solution

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 $f (x) = x^{4} + 2 x$ 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 $[- 1, 1]$ .

Hide solution and development