Answer:
The graph of the polynomial function f(x) = x^4 + x^3 - 2x^2 will depend on the behavior of the function as x approaches infinity and negative infinity, as well as the location and behavior of any local extrema.
To determine the behavior of the function as x approaches infinity and negative infinity, we can look at the leading term of the polynomial, which is x^4. As x becomes very large (either positive or negative), the x^4 term will dominate the expression, and f(x) will become very large in magnitude. Therefore, the graph of the function will approach positive or negative infinity as x approaches infinity or negative infinity, respectively.
To find any local extrema, we can take the derivative of the function and set it equal to zero:
f(x) = x^4 + x^3 - 2x^2
f'(x) = 4x^3 + 3x^2 - 4x
Setting f'(x) equal to zero, we get:
4x(x^2 + 3/4x - 1) = 0
The solutions to this equation are x = 0 and the roots of the quadratic expression x^2 + 3/4x - 1. Using the quadratic formula, we can find these roots to be:
x = (-3 ± sqrt(33))/8
Therefore, the critical points of the function are x = 0 and x = (-3 ± sqrt(33))/8.
To determine the behavior of the function near each critical point, we can use the second derivative test. Taking the second derivative of f(x), we get:
f''(x) = 12x^2 + 6x - 4
Evaluating f''(0), we get:
f''(0) = -4
Since f''(0) is negative, we know that x = 0 is a local maximum of the function.
Evaluating f''((-3 + sqrt(33))/8), we get:
f''((-3 + sqrt(33))/8) = 11 + 3 sqrt(33)/2
Since f''((-3 + sqrt(33))/8) is positive, we know that x = (-3 + sqrt(33))/8 is a local minimum of the function.
Evaluating f''((-3 - sqrt(33))/8), we get:
f''((-3 - sqrt(33))/8) = 11 - 3 sqrt(33)/2
Since f''((-3 - sqrt(33))/8) is also positive, we know that x = (-3 - sqrt(33))/8 is another local minimum of the function.
Based on this information, we can sketch the graph of the function as follows:
- As x approaches negative infinity, the graph of the function approaches negative infinity.
- The function has a local maximum at x = 0.
- The function has two local minima at x = (-3 ± sqrt(33))/8.
- As x approaches infinity, the graph of the function approaches positive infinity.
Therefore, the statement that describes the graph of this polynomial function is: "The graph of the function has a local maximum at x = 0 and two local minima at x = (-3 ± sqrt(33))/8. As x approaches infinity or negative infinity, the graph of the function approaches positive or negative infinity, respectively."