SOLUTION: function: y= x^4-8x^3-16x+5 a. Identify range b. find x-intercept (you can solve this using the intermediate value theorem to estimate where the roots will be, I'm just not sure

Question 1180565: function: y= x^4-8x^3-16x+5
a. Identify range
b. find x-intercept (you can solve this using the intermediate value theorem to estimate where the roots will be, I'm just not sure how to do that exactly)
c. find critical numbers
d. points of extreme with the first derivative test for the interval of increasing and decreasing testing
- please show the work, I did some work on my own but I think I messed up in some steps. thank you very much.
Answer by Boreal(15235) (Show Source): You can put this solution on YOUR website!
I would graph it first. Clearly +oo is the upper bound

We know that when x=0 y=5, and as x increases from 0, the function will be decreasing, since subtracting the cube of a small positive number and the first power of a small positive number will both be larger than the adding the fourth power of that number.
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So one root will be between 0 and 1, closer to 0
when x=8, the x^4 and -8x^3 terms will cancel, making the function -123. Therefore, the next root will be somewhat larger than 8 and well below 9 (closer to 8).
The other two roots will be complex.
The critical values are at x=0.2996 and x=8.227. The function is continuous.
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The derivative is y'=4x^3-24x^2-16
set that equal to 0 and divide by 4, and the is x^3-6x^2-4=0
this is negative when x=6 and positive when x=7, so the local minimum is when x is in between those two points.
x=6.11
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So the range is (-523.8679, +oo)

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