Respuesta :
Answer:
Step-by-step explanation:
f(x) = x⁵ – 8x⁴ + 16x³
As x approaches +∞, the highest term, x⁵, approaches +∞.
As x approaches -∞, x⁵ approaches -∞ (a negative number raised to an odd exponent is also negative).
Now let's factor:
f(x) = x³ (x² – 8x + 16)
f(x) = x³ (x – 4)²
f(x) has roots at x=0 and x=4. x=4 is a repeated root (because it's squared), so the graph touches the x-axis but does not cross at x=4.
The graph crosses the x-axis at x=0.
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.
Read more at; https://brainly.com/question/14962400
