Final answer:
To find the equations of the lines tangent to y = 9 - x³, we need to find the derivative of the function and then find the point of tangency and the slope of the tangent line at that point. By substituting different x values, we can find the equations of all the tangent lines: y = 0, y = 9, y = 3x + 6, and y = -x + 9.
Step-by-step explanation:
The equation y = 9 - x³ represents a cubic function. To find the equations of the lines tangent to this curve, we need to find the derivative of the function.
Let's find the derivative of y = 9 - x³:
Take the derivative of the constant term 9, which is 0.
Take the derivative of the term -x³ using the power rule. The derivative of x³ is 3x². Since the original term is negative, the derivative is -3x².
Therefore, the derivative of y = 9 - x³ is dy/dx = 0 - 3x² = -3x².
Now, we can find the equations of the tangent lines by finding the point of tangency and the slope of the tangent line at that point.
For each value of x, we substitute it into the derivative -3x² to find the slope of the tangent line at that point.
Then we substitute the x value into the original equation y = 9 - x³ to find the corresponding y value.
The equation of the tangent line is then y - y₁ = m(x - x₁), where (x₁, y₁) is a point on the curve and m is the slope of the tangent line.
By substituting different x values, we can find the equations of all the tangent lines:
A) y = 0
B) y = 9
C) y = 3x + 6
D) y = -x + 9