If f(x) = |(x2 - 6)(x2 + 2)|, how many numbers in the interval 1 ≤ x ≤ 2 satisfy the conclusion of the mean value theorem?
none
one
two
three

Question

If f(x) = |(x2 - 6)(x2 + 2)|, how many numbers in the interval 1 ≤ x ≤ 2 satisfy the conclusion of the mean value theorem? 
none
one
two
three

sbentleyaz · Answer

@3mar

3mar · Answer

Sorry I will be back within 45 min Salam!

tkhunny · Answer

The slope between the two endpoints needs to be reproduced by the slope of the tangent line inside the interval.  You should start by finding the slope between the two endpoints.

sbentleyaz · Answer

I got the slope as 3

sbentleyaz · Answer

im not sure if thats correct

tkhunny · Answer

Give it another go. It's not +3, it's ...

sbentleyaz · Answer

-3?

tkhunny · Answer

That's the spirit. Now, MVT wants us to find where the value of the 1st derivative is -3. Accordding to MVT, there MUST be one such place. There may be more. You'll need to find the first derivative and see where it takes on the value -3.

sbentleyaz · Answer

would the derivative be 4x^2?

tkhunny · Answer

It's far more complex than that.

There's an interesting way to get rid fo the Absolute Values,in this case.

The stuff inside is always negative on our interval.  Just discard the absolute values and make the expression negative.

f(x) = -(x2 - 6)(x2 + 2)

Now, try that derivative, again.

sbentleyaz · Answer

i got 
$$-4x ^{3}+8x$$

tkhunny · Answer

Did you apply the WHOLE Multiplication rule?

sbentleyaz · Answer

i think

tkhunny · Answer

...because if you did, you did it EXACTLY correctly.

Okay, where is that thing -3?

sbentleyaz · Answer

do you substitute in -3 for x?

tkhunny · Answer

No, other way.  We need a value of x so that the entire expression is -3.

sbentleyaz · Answer

so set it equal to -3

tkhunny · Answer

Right.  Youmay with to notice, just before you do that , that it takes on zero at x = 0 and x = +/- sqrt(2).  Only one of those, x = sqrt(2) is in our interval.  Just something to notice.  Okay, look for -3.

sbentleyaz · Answer

wait if only one falls in it doesn't that mean theres only one possible answer

tkhunny · Answer

Sorry, that may have been distracting.  I was just thinking about f'(x) = 0, since all three solutions were so obvious.  You're still looking for f'(x) = -3.

tkhunny · Answer

Anyway, that's all you need.  This problem doesn't care WHAT it is, it just wants to know how many.  Since the slope of this derivative is always negative, it never turns around, so there is only one.

sbentleyaz · Answer

so technically there are 3 but for the limitations of the problem there would only be 1?

tkhunny · Answer

No, technically, there is only one.