Question

Gary recently lost his graphing calculator. However, he has spent his last $100 on a sweater from the Smithsonian so he is not able to buy another. This is quite unfortunate, since Gary has to determine the real root to the equation y = x3 + 2x − 4 immediately to gain access to the Smithsonian Fancy Sweater Club. Gary grabs a paper and pencil and is able to determine an interval that the root falls between, so the club members take pity on him and grant him access. In which of the following intervals does Gary determine that the root of the equation falls?
A. [-1,0] B. [0,1] C. [1,2] D. [2,3]

Answer 1

[1,2]

Answer 2

How? It mentions something about the intermediate value theorem, but I don't get it? From Descartes Rule of Signs and Rational Root Theorem, all I could figure out was [1, 4], but I don't know the intermediate value theorem.

Answer 3

The f(1) is negative
The f(2) is positive
Therefore the graph must cross the x axis between 1 and 2 and if it crosses the x axis, that is a root.

Answer 4

Ok. So let me see if I get it. Since the end result of plugging in a x-value gives you a corresponding y-value, if the sign switched, it must cross the x-axis because the positive and negative y-values are separated by the x-axis?

Answer 5

Yes. Think if you were graphing the function. You would graph (1,-1) and (2,8). When you started to actually the graph, how could you get from (1,-1) to (2,8) unless you crossed the x axis?

Answer 6

Ok. Thank you so much!

Answer 7

*actually draw the graph

Answer 8

yw