Final answer:
To find the critical points of the function f(x) = (3x^2) / (5x^2 - 2x + 10), we need to find the values of x where the derivative of f(x) equals zero. Once we find the values of x that make the derivative equal to zero, we can substitute them back into the original function f(x) to find the corresponding y-values for the critical points.
Step-by-step explanation:
To find the critical points of the function f(x) = (3x^2) / (5x^2 - 2x + 10), we need to find the values of x where the derivative of f(x) equals zero. The critical points are the x-values that make the derivative of f(x) equal to zero or undefined. To find these points, we need to calculate the derivative of f(x) and then solve for x.
To calculate the derivative of f(x), we can use the quotient rule: [ (d/dx)(3x^2) * (5x^2 - 2x + 10) - (3x^2) * (d/dx)(5x^2 - 2x + 10) ] / (5x^2 - 2x + 10)^2.
Simplifying this expression, we get (18x^3 - 8x^2 - 80x + 80) / (5x^2 - 2x + 10)^2 = 0.
This equation can be solved for x using algebraic techniques such as factoring or the quadratic formula. Once we find the values of x that make the derivative equal to zero, we can substitute them back into the original function f(x) to find the corresponding y-values for the critical points.