Question

What is the end behavior of the graph of the polynomial function `f(x) = 3x^6 + 30x^5 + 75x^4`?

Answer 1

As `x to+-oo, y to oo`. Turning points at `x - 0, -10/3 and -5`.
Illustrative Socratic graph is inserted.

Explanation:

`y = f(x) = 3x^4(x+5)^2` >=0.

zeros: `0, 0, 0, 0, -5 and -5`

x-intercept ( y = 0 ): x = -5.

Local mini/max turning points ( y' = 0 ): 0 0, 0, -10/3 and -5.-,

As `x to+-oo, y to oo`.

graph{3x^6+30x^5+75x^4 [-9.03, 9.14, -1100,1100]}