Does the function has a zero in the given interval? Explain why.

f(x) = x^4 - 5 on [-1, 2]

Question

Does the function has a zero in the given interval? Explain why.

f(x) = x^4 - 5 on [-1, 2]

thesmartone · Answer

This is a question on IVT
Intermediate value theorem

thesmartone · Answer

So when x = -1
f(-1) = (-1)^4 - 5 = 1 - 5 = -4
f(-1) = -4

When x = 2
f(2) = (2)^4 - 5 = 8 - 5 = 3
f(2) = 3

thesmartone · Answer

So by IVT there will be some x-value between -1 and 2 that will cross the x-axis.

Right? o:

mathmale · Answer

Hello!
I'd suggest you experiment a bit.  First, sketch y=x^4.  Doesn't have to be a fancy sketch.
Next, 

Sketch z=x^4-5 on the same set of axes.  Does this "new" function seem to have any roots?  From your graph alone, you may be able to determine whether one or both of these roots lies within the interval [-1,2].

mathmale · Answer

Your approach, using the IMV, is better than mine.  Good for you.

3mar · Answer

Here is your function graphed.

You decide whatever has a root (roots) or not.

thesmartone · Answer

Yes, it does https://www.desmos.com/calculator/ljdqge2hew

thesmartone · Answer

Haha, yes. I'm learning Calc 1 and we just learned about IVT so they want us to apply that in solving it :)

3mar · Answer

here

thesmartone · Answer

Thank you 3mar, but we aren't solving for the zero. We're simply trying to prove that there will be a certain y-value in a given range of x-values. :)

3mar · Answer

Ok

3mar · Answer

I just wanted to help

thesmartone · Answer

Yes, that's alright :)