Find all critical points of the function $\newline$ $f(x)=\frac{5x^{2}}{5x^{2}-2x+10}.$ $\newline$ (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.)

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Q. Find all critical points of the function

\newline

f(x)=\frac{5x^{2}}{5x^{2}-2x+10}.

\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.)

Calculate Derivative: To find the critical points of the function $f(x) = \frac{5x^2}{5x^2 - 2x + 10}$ , we need to find the values of $x$ where the derivative $f'(x)$ is either zero or undefined.
Apply Quotient Rule: First, we calculate the derivative of $f(x)$ using the quotient rule, which states that if $f(x) = \frac{g(x)}{h(x)}$ , then $f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{(h(x))^2}$ .
Simplify Numerator: Let $g(x) = 5x^2$ and $h(x) = 5x^2 - 2x + 10$ . Then $g'(x) = 10x$ and $h'(x) = 10x - 2$ .
Set Equal to Zero: Using the quotient rule, $f'(x) = \frac{10x \cdot (5x^2 - 2x + 10) - 5x^2 \cdot (10x - 2)}{(5x^2 - 2x + 10)^2}$ .
Factor and Solve: Simplify the numerator of $f'(x)$ : $10x \cdot (5x^2 - 2x + 10) - 5x^2 \cdot (10x - 2) = 50x^3 - 20x^2 + 100x - 50x^3 + 10x^2$ .
Factor and Solve: Simplify the numerator of $f'(x)$ : $10x \times (5x^2 - 2x + 10) - 5x^2 \times (10x - 2) = 50x^3 - 20x^2 + 100x - 50x^3 + 10x^2$ .Further simplification of the numerator gives us $-10x^2 + 100x$ .
Factor and Solve: Simplify the numerator of $f'(x)$ : $10x \times (5x^2 - 2x + 10) - 5x^2 \times (10x - 2) = 50x^3 - 20x^2 + 100x - 50x^3 + 10x^2$ .Further simplification of the numerator gives us $-10x^2 + 100x$ .Now we have $f'(x) = \frac{-10x^2 + 100x}{(5x^2 - 2x + 10)^2}$ .
Factor and Solve: Simplify the numerator of $f'(x)$ : $10x \times (5x^2 - 2x + 10) - 5x^2 \times (10x - 2) = 50x^3 - 20x^2 + 100x - 50x^3 + 10x^2$ . Further simplification of the numerator gives us $-10x^2 + 100x$ . Now we have $f'(x) = \frac{-10x^2 + 100x}{(5x^2 - 2x + 10)^2}$ . To find the critical points, we set the numerator equal to zero because the denominator cannot be zero (it's always positive due to the square and the constant term $10$ ).
Factor and Solve: Simplify the numerator of $f'(x)$ : $10x \times (5x^2 - 2x + 10) - 5x^2 \times (10x - 2) = 50x^3 - 20x^2 + 100x - 50x^3 + 10x^2$ . Further simplification of the numerator gives us $-10x^2 + 100x$ . Now we have $f'(x) = \frac{-10x^2 + 100x}{(5x^2 - 2x + 10)^2}$ . To find the critical points, we set the numerator equal to zero because the denominator cannot be zero (it's always positive due to the square and the constant term $10$ ). Solve $-10x^2 + 100x = 0$ for $x$ . We can factor out a $10x$ : $10x(-x + 10) = 0$ .
Factor and Solve: Simplify the numerator of $f'(x)$ : $10x \times (5x^2 - 2x + 10) - 5x^2 \times (10x - 2) = 50x^3 - 20x^2 + 100x - 50x^3 + 10x^2$ .Further simplification of the numerator gives us $-10x^2 + 100x$ .Now we have $f'(x) = \frac{-10x^2 + 100x}{(5x^2 - 2x + 10)^2}$ .To find the critical points, we set the numerator equal to zero because the denominator cannot be zero (it's always positive due to the square and the constant term $10$ ).Solve $-10x^2 + 100x = 0$ for $x$ . We can factor out a $10x$ : $10x(-x + 10) = 0$ .Setting each factor equal to zero gives us two solutions: $x = 0$ and $10x \times (5x^2 - 2x + 10) - 5x^2 \times (10x - 2) = 50x^3 - 20x^2 + 100x - 50x^3 + 10x^2$ $0$ .
Factor and Solve: Simplify the numerator of $f'(x)$ : $10x \cdot (5x^2 - 2x + 10) - 5x^2 \cdot (10x - 2) = 50x^3 - 20x^2 + 100x - 50x^3 + 10x^2$ .Further simplification of the numerator gives us $-10x^2 + 100x$ .Now we have $f'(x) = \frac{-10x^2 + 100x}{(5x^2 - 2x + 10)^2}$ .To find the critical points, we set the numerator equal to zero because the denominator cannot be zero (it's always positive due to the square and the constant term $10$ ).Solve $-10x^2 + 100x = 0$ for $x$ . We can factor out a $10x$ : $10x(-x + 10) = 0$ .Setting each factor equal to zero gives us two solutions: $x = 0$ and $10x \cdot (5x^2 - 2x + 10) - 5x^2 \cdot (10x - 2) = 50x^3 - 20x^2 + 100x - 50x^3 + 10x^2$ $0$ .Therefore, the critical points of the function are $x = 0$ and $10x \cdot (5x^2 - 2x + 10) - 5x^2 \cdot (10x - 2) = 50x^3 - 20x^2 + 100x - 50x^3 + 10x^2$ $0$ .