Final Answer:
The maximum value of the equation y = -x^6 + 2x^3 + 4x^4 - 12 is -8.
Step-by-step explanation:
The given equation y = -x^6 + 2x^3 + 4x^4 - 12 is a polynomial equation of degree 6. The highest degree of the equation is 6, thus the equation has a maximum of 6 turning points. However, the equation has only 4 turning points, which are the points where the gradient of the equation equals 0. To find the maximum value of the equation, we need to calculate the value of x when the gradient of the equation equals 0.
The gradient of the equation is calculated by differentiating the equation with respect to x. The derivative of the equation is -6x^5 + 6x^2 + 16x^3. The turning points of the equation are found when the derivative of the equation is equal to 0. Thus, the turning points of the equation are found when -6x^5 + 6x^2 + 16x^3 = 0. This is a polynomial equation of degree 5, which can be solved using the quadratic formula.
The turning points of the equation are x = 0, x = ±2/3 and x = ±1/2. To find the maximum of the equation, we need to substitute each of the turning points into the equation and calculate the value of y. At x = 0, the value of y = -12. At x = ±2/3, the value of y = -8. At x = ±1/2, the value of y = -7. Thus, the maximum value of the equation y = -x^6 + 2x^3 + 4x^4 - 12 is -8.