Question

The value 4 is an upper bound for the zeros of the function shown below.
f(x)=4x^3-12x^2-x+15

A. True
B. False

Answer 1

true

Answer 2

Why is that?

Answer 3

f(x) = 4x^3 – 12x^2 – x + 15.
There are two sign changes, hence, by Descart’s rule of signs, there exist 2 or 0 positive zeros
f(-x) = -4x^3 – 12x^2 + x + 15.
There is only one sign change in this case, hence, by Descart’s rule of signs, there is exactly one
negative root.
The leading coefficient is 4, so we must divide all terms by 4
x^3 – 3x^2 – 0.25x + 3.75
The coefficients are 1; -3; -0.25; 3.75.
Drop the leading coefficient and remove any minus signs: 3; 0.25; 3.75
Bound 1: the largest value is 3.75. Plus 1 will be 4.75.
Bound 2: adding all values is 3+0.25+3.75=7. The sum of all values is greater than 1.
The smallest “bounds” value (4.75) is our answer. All roots are within plus or minus of that.

Answer 4

its falsee

Answer 5

@asiannerd

Answer 6

Wow thanks!

Answer 7

np