Question

Use the intermediate value theorem to show that there is a root of the equation 2x^3 +x^2+2=0 in the interval [-2,-1]

Answer 1

when you plug in x = -2, what do you get?

Answer 2

-14

Answer 3

nope i mean -10

Answer 4

close, but not quite

Answer 5

good

Answer 6

now plug in x = -1

Answer 7

=1

Answer 8

so because y changes from -10 to 1, this means that y must be equal to 0 at least once in this interval (assuming the function is continuous)

Answer 9

there's no way for a continuous graph to go from a negative y value then jump to a positive y value without passing through y = 0 first

so that proves there is at least one root in this interval

Answer 10

is there a mathematic way to show that answer or just explaining it?

Answer 11

or no?

Answer 12

well the mathematical way would be something like this

Let f(x) be a continuous function on the interval [a,b] where a and b are two numbers (b > a). Let's introduce the number w such that w is between f(a) and f(b), ie

f(a) < w < f(b)

according to the intermediate value theorem, there is at least one value c such that

f(c) = w

Answer 13

It's explained in more detail here (with a graph too)

http://www.mathsisfun.com/algebra/intermediate-value-theorem.html

Answer 14

so the idea is this

let's make f(a) some negative number
let's make f(b) some positive number

since f(x) is continuous, we can let w = 0 and by the intermediate value theorem we know there is at least one value of c such that

f(c) = 0

Answer 15

this works because f(a) < w < f(b) is true

Answer 16

hopefully that makes sense, if not, then just stick to the english explanation

Answer 17

me personally, I like to think of it as a river that's infinitely long like this

Answer 18

If you're in the negative region or below the river, then the only way to get to the positive region or above the river is to cross the river itself

The river represents the x axis. Crossing the x axis is where roots or zeros occur

Answer 19

yea i still don't get how to write it mathematically but i know that is what my teacher willw ant

Answer 20

i get that you have to cross the x axis i just don't know how to show it mathematically

Answer 21

try your best to translate this type of explanation into mathematical terms

Answer 22

you could try this

when x = -2, y = -10
when x = -1, y = 1

y = 2x^3 + x^2 + 2 is a continuous function for all real numbers x

so there are no breaks, jumps, or gaps in the graph, which means that the only way for y to change from a negative to a positive is that y must be 0 at some point (at least one time). So that means 2x^3 +x^2+2 has at least one root.

Answer 23

okay i willl try that, thank you!

Answer 24

you're welcome