Find all critical points of the function f(x)=(3x^(2))/(7x^(2)-2x+2). (Use symbolic notation and fractions where needed. Give your answer in the form of a comma separated list. If the function does not have any critical points, enter DNE.) critical points:

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Find all critical points of the function  f(x)=(3x^(2))/(7x^(2)-2x+2). (Use symbolic notation and fractions where needed. Give your answer in the form of a comma separated list. If the function does not have any critical points, enter DNE.) critical points:

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Calculate Derivative: To find the critical points of the function f(x) = (3x^2) / (7x^2 - 2x + 2), we need to find the values of x where the derivative f'(x) is zero or undefined.Apply Quotient Rule: First, we calculate the derivative of f(x) using the quotient rule, which states that if f(x) = g(x)/h(x), then f'(x) = (g'(x)h(x) - g(x)h'(x)) / (h(x))^2. Here, g(x) = 3x^2 and h(x) = 7x^2 - 2x + 2.Simplify Numerator: Compute the derivative of g(x) = 3x^2, which is g'(x) = 6x.Find Critical Points: Compute the derivative of h(x) = 7x^2 - 2x + 2, which is h'(x) = 14x - 2.Factor Common Term: Apply the quotient rule to find f'(x): f'(x) = (6x * (7x^2 - 2x + 2) - 3x^2 * (14x - 2)) / (7x^2 - 2x + 2)^2.Solve for x: Simplify the numerator of f'(x): Numerator = 6x * (7x^2 - 2x + 2) - 3x^2 * (14x - 2) = 42x^3 - 12x^2 + 12x - 42x^3 + 6x^2 = -12x^2 + 12x + 6x^2 = -6x^2 + 12x.Check Denominator: Set the simplified numerator equal to zero to find the critical points: -6x^2 + 12x = 0.Identify Critical Points: Factor out the common term: -6x(x - 2) = 0.Set each factor equal to zero and solve for x: -6x = 0 gives x = 0. x - 2 = 0 gives x = 2.Check if the denominator of f'(x) becomes zero at x = 0 or x = 2, which would make the derivative undefined: (7x^2 - 2x + 2)^2 at x = 0 is (2)^2, which is not zero. (7x^2 - 2x + 2)^2 at x = 2 is (7(2)^2 - 2(2) + 2)^2, which is (28 - 4 + 2)^2 = (26)^2, which is not zero.Since the denominator of f'(x) does not become zero at x = 0 or x = 2, the critical points are x = 0 and x = 2.

Find all critical points of the function $\newline$ $f(x)=\frac{3x^{2}}{7x^{2}-2x+2}.$ $\newline$ (Use symbolic notation and fractions where needed. Give your answer in the form of a comma separated list. If the function does not have any critical points, enter DNE.) $\newline$ critical points:

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