Answer: f(x) = x⁵ – 8x⁴ + 16x³
Does not cross the x-axis at x = 4.
Crosses the x-axis at x = 0
This is about the end behavior of a graph of a function at the end of the x-axis.
We are given the function;
f(x) = x⁵ - 8x⁴ + 16x³
A) As x approaches negative infinity -∞, x⁵ will also approach negative infinity -∞. This is because when we raise a negative number to the power of an odd number, the result remains negative.
B) As x approaches positive infinity +∞, x⁵ will also approach positive infinity -∞. This is because when we raise a positive number to the power of an odd number, the result remains positive.
Let's now find the roots of this function;
f(x) = x⁵ - 8x⁴ + 16x³
Let's factorize it first to get;
f(x) = x³(x² – 8x + 16)(x² – 8x + 16) is a perfect square trinomial and can be expressed as (x – 4)(x - 4).
Thus;f(x) = x³ (x – 4)(x - 4)
C) Since we have found the factorized form to be;
f(x) = x³ (x – 4)(x - 4)
The roots are at f(x) = 0;
The roots are; x³ = 0; (x – 4) = 0 ; (x - 4) = 0
This means the roots of f(x) are; x=0 and x=4. x = 4
This means the graph has a repeated root and so it will touch the x-axis but not at the repeated root of x=4.
D) Since it 0 is a root and it does not cross at x = 4, the graph will cross at x = 0.
