Solved: "Use the Intermediate Value Theorem to show that..."

Khaleesi Herbert

Answered question

2021-09-15

Use the Intermediate Value Theorem to show that there is a root of the given equation in the specified interval.

x^{4} + x - 3 = 0,

(1,2)

Answer & Explanation

SchepperJ

Skilled2021-09-16Added 96 answers

Step 1
Let $f (x) = x^{4} + x - 3$ for all $x \in ⟨ 1, 2 ⟩$ . We are looking for a number $c \in [1, 2]$ such that $f (c) = 0$ . Function f is continuous on the closed interval $[1, 2]$ so we can use Intermediate Value Theorem. We take $a = 1, b = 2, N = 0$ in intermediate Value Theorem. We have:
$f (1) = 1^{4} + 1 - 3 = - 1 < 0$
$f (2) = 2^{2} + 2 - 3 = 15 > 0$
Thus $f (1) < 0 < f (2)$ , that is , $N = 0$ is a number between $f (1)$ and $f (2)$ so the mentioned theorem says there is a number c between 1 and 2 such that $f (c) = 0$ . We conclude that initial equation has at least one root $c \in [1, 2]$
$f (1) = 1^{4} + 1 - 3 = - 1 < 0$
$f (2) = 2^{4} + 2 - 3 = 15 > 0$
so there exist $c \in ⟨ 1, 2 ⟩$ such that $f (x) = 0$

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