Final answer:
To find F'(3), use the quotient rule to find the derivative of F(x), substitute x = 3 into F'(x) to find F'(3), and simplify the expression. The value of F'(3) is -8/9. To find the equation of the tangent line to the curve y = 12x/(3 + x²) at the point (3, 3), use the slope-intercept form of a linear equation and substitute the slope and the coordinates of the point into the equation. The equation of the tangent line is y = (-8/9)x + 93/9.
Step-by-step explanation:
To find F'(3), we first need to find the derivative of F(x). We can do this by using the quotient rule, which states that the derivative of f(x) = g(x) / h(x) is given by f'(x) = (g'(x) * h(x) - g(x) * h'(x)) / (h(x))². Applying this rule to the function F(x) = 12x / (3 + x²), we have:
F'(x) = ((12 * (3 + x²)) - (12x * (2x))) / ((3 + x²)²)
Now, substitute x = 3 into F'(x) to find F'(3):
F'(3) = ((12 * (3 + 3²)) - (12 * 3 * (2 * 3))) / ((3 + 3^2)²)
F'(3) = ((12 * (3 + 9)) - (36 * 6)) / ((3 + 9)²)
Performing the calculations, we get F'(3) = -8/9 as the value of F'(3).
Now, to find the equation of the tangent line to the curve y = 12x / (3 + x²) at the point (3, 3), we can use the slope-intercept form of a linear equation, which is y = mx + b, where m is the slope and b is the y-intercept. Since the slope of the tangent line is given by F'(3) (which we found to be -8/9), and the point (3, 3) lies on the line, we can substitute these values into the equation to get:
y = (-8/9)x + b
Using the point (3, 3), we can solve for b:
3 = (-8/9)(3) + b
Simplifying, we find b = 93/9. Therefore, the equation of the tangent line to the curve at the point (3, 3) is y = (-8/9)x + 93/9.