Problems from Property of Darboux (theorem of the intermediate value)

Say if the following equations have any solution using the Darboux property.

a) $x^{2} = 1$

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

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

See development and solution

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 $f ([1, 2]) = [2.7182 \dots, 6.69 \dots]$ 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 $f (x) = x^{4} + 2 x$ 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 $[- 1, 1]$ .

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