To determine the domain and range of the polynomial \( y = -4x^4 + 4x^3 + 35x^2 - 9 \), we can analyze it as follows:
Domain: The domain of any polynomial function is all real numbers since polynomials are defined for every real number \( x \). Hence, the domain is: \[ \text{Domain: } (-\infty, \infty) \]
Range: To find the range, we need to look for the maximum or minimum values of the polynomial. Since the leading coefficient is negative (\(-4\)), the graph of the polynomial opens downward, indicating it will have a maximum point.
We can find the critical points by taking the derivative of the polynomial and setting it equal to zero:
-
Calculate the derivative: \[ y' = -16x^3 + 12x^2 + 70x \]
-
Set the derivative to zero to find critical points: \[ -16x^3 + 12x^2 + 70x = 0 \] Factor out the common term \( x \): \[ x(-16x^2 + 12x + 70) = 0 \] This gives one critical point at \( x = 0 \).
-
For the quadratic \( -16x^2 + 12x + 70 \), we can use the quadratic formula to find its roots, if needed: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-12 \pm \sqrt{12^2 - 4(-16)(70)}}{2(-16)} \] Calculate the discriminant: \[ = 144 + 4480 = 4624 \] Now find the roots: \[ x = \frac{-12 \pm 68}{-32} \] From this, we would find two critical points.
-
After determining all critical points, evaluate the function \( y \) at all critical points as well as at the endpoints where \( x \to \pm \infty \) (as this will approach \(-\infty\)).
Given that the maximum y-value of this polynomial occurs at its highest local maximum point, we can numerically find or estimate that this maximum value is approximately \( 116 \).
Thus, the range of the polynomial will be: \[ \text{Range: } (-\infty, 116] \]
Final Answer:
- Domain: \( (-\infty, \infty) \)
- Range: \( (-\infty, 116] \)
Based on the responses provided in the question, the correct option is: Domain: \((-∞, ∞)\); Range: \((−∞, 116]\).
1 answer