Use the intermediate value theorem to show that there is a root of the equation $x^5-2x^4-x-3=0$ in the interval $(2,3)$ ?

Question

Use the intermediate value theorem to show that there is a root of the equation $x^5-2x^4-x-3=0$ in the interval $(2,3)$ ?

1 Answer

Impact of this question

4758 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2018-05-23T10:43:53

See below for proof.

Explanation:

If $f(x)=x^5-2x^4-x-3$
then
$color(white)("XXX")f(color(blue)2)=color(blue)2^5-2 * color(blue)2^4-color(blue)2-3=color(red)(-5)$
and
$color(white)("XXX")f(color(blue)3)=color(blue)3^5-2 * color(blue)3^4-color(blue)3-3=243-162-3-3=color(red)(+75)$

Since $f(x)$ is a standard polynomial function, it is continuous.

Therefore, based on the intermediate value theorem, for any value, $color(magenta)k$ , between $color(red)(-5)$ and $color(red)(+75)$ , there exists some $color(lime)(hatx)$ between $color(blue)2$ and $color(blue)3$ for which $f(color(lime)(hatx))=color(magenta)k$

Since $color(magenta)0$ is such a value,
there exists some value $color(lime)(hatx) in [color(blue)2,color(blue)3]$ such that $f(color(lime)(hatx))=color(magenta)0$

Science

Math

Humanities

... and beyond

Use the intermediate value theorem to show that there is a root of the equation $x^5-2x^4-x-3=0$ in the interval $(2,3)$ ?

1 Answer

Explanation:

Related questions

Impact of this question

Science

Math

Humanities

... and beyond

Use the intermediate value theorem to show that there is a root of the equation x^5-2x^4-x-3=0x^5-2x^4-x-3=0 in the interval (2,3)(2,3)?

1 Answer

Explanation:

Related questions

Impact of this question

Use the intermediate value theorem to show that there is a root of the equation $x^5-2x^4-x-3=0$ in the interval $(2,3)$ ?