f(x)=x^2(x^2-2)(x+2)

Determine the end behavior, find the power function that the graph of f resembles for large values of |x|
Find the maximum number of turning points

If we expand we get

f(x) = x^5 ..... where x^5 will be the dominating term
for positive x's, obviously f(large +x) ---> + ∞
for negative x's, f(large -x) ----> - ∞

To determine the end behavior, we need to look at the highest-degree term in the polynomial function.

In the given function f(x) = x^2(x^2-2)(x+2), the highest-degree term is x^2.

For large values of |x|, x^2 will dominate the behavior of the function. So, the end behavior of f(x) as x approaches positive or negative infinity is the same as the end behavior of the function g(x) = x^2.

For x approaching positive infinity (x → +∞), g(x) = x^2 will tend to positive infinity.

For x approaching negative infinity (x → -∞), g(x) = x^2 will also tend to positive infinity because squaring a negative number gives a positive result.

Therefore, the end behavior of f(x) as x approaches positive or negative infinity is that it tends to positive infinity.

Now, let's find the power function that the graph of f resembles for large values of |x|.

As mentioned earlier, the end behavior of f(x) is the same as the end behavior of g(x) = x^2. So, a power function that the graph of f(x) resembles for large values of |x| is h(x) = x^2.

Finally, let's find the maximum number of turning points in the graph of f(x).

The graph of a polynomial function can have turning points where the derivative of the function is equal to zero.

To find the number of turning points, we need to determine the number of real roots the derivative of f(x) = x^2(x^2-2)(x+2) has.

Taking the derivative of f(x), we get:
f'(x) = 4x^3 - 8x^2 + 2x - 4

Now, we can find the roots of f'(x) by setting it equal to zero and solving for x.

4x^3 - 8x^2 + 2x - 4 = 0

Unfortunately, finding the exact roots of this cubic equation may not be easily achievable. However, we can use numerical methods or a graphing calculator to estimate the number of real roots.

From the shape of the graph of f'(x), we can see that there might be two turning points.

Therefore, the maximum number of turning points in the graph of f(x) is 2.

To determine the end behavior of the function f(x) = x^2(x^2-2)(x+2), you can look at the highest-degree term in the polynomial. In this case, the highest-degree term is x^2(x^2-2)(x+2).

For large values of |x| (as |x| approaches infinity), the dominant term is x^2. As x^2 increases, the entire expression x^2(x^2-2)(x+2) will also increase. Therefore, the end behavior of the function is that it approaches positive infinity as x approaches positive infinity, and it approaches negative infinity as x approaches negative infinity.

Now, let's find the power function that the graph of f(x) resembles for large values of |x|. Since the dominant term is x^2, the power function that the graph of f(x) resembles for large values of |x| is f(x) ≈ x^2.

To find the maximum number of turning points, you can look at the degree of the polynomial. The function f(x) = x^2(x^2-2)(x+2) is a fourth-degree polynomial because the highest power of x is x^2.

A polynomial of degree n can have at most n-1 turning points. Therefore, a fourth-degree polynomial can have at most 3 turning points. Hence, the maximum number of turning points for the function f(x) = x^2(x^2-2)(x+2) is 3.