What is the end behavior of the graph of f(x) = x⁵ - 8x⁴ + 16x³?

Solution:

Given, f(x) = x⁵ - 8x⁴ + 16x³

We have to find the end behavior of the graph.

1) As x approaches -∞, x⁵ will also approach -∞. This is because when we raise a negative number to the power of an odd number, the result remains negative.

2) As x approaches +∞, x⁵ will also approach +∞. This is because when we raise a positive number to the power of an odd number, the result remains positive.

3) Finding the roots of the given function,

f(x) = x⁵ - 8x⁴ + 16x³

Taking x³ as common,

x³(x² - 8x + 16) = 0

x³(x² - 4x - 4x + 16) = 0

x³(x(x - 4) - 4(x - 4)) = 0

x³(x - 4)(x - 4) = 0

x³ = 0

So, x = 0

x - 4 = 0

x = 4

So, the roots are x = 0, and x = 4.

This implies that the graph has a repeated root and so it will touch the x-axis but not at the repeated root of x = 4.

4) Since 0 is a root and it does not cross at x = 4, the graph will cross at x = 0.

Therefore, the graph does not cross the x-axis at x = 4 but crosses the x-axis at x = 0.

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What is the end behavior of the graph of f(x) = x⁵ - 8x⁴ + 16x³?

Summary:

The end behavior of the graph of f(x) = x⁵ - 8x⁴ + 16x³ is that  as x → -∞, x⁵ → -∞ and  as x → ∞, x⁵ → ∞ and the graph does not cross the x-axis at x = 4 but crosses the x-axis at x = 0.