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.) $\newline$ critical points:

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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.)

\newline

critical points:

Find Critical Points: To find the critical points of the function, we need to find the values of $x$ where the derivative of the function is $0$ or undefined.
Calculate Derivative: First, we find the derivative of the function $f(x)$ using the quotient rule, which states that if $f(x) = \frac{g(x)}{h(x)}$ , then $f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{(h(x))^2}$ .
Apply Quotient Rule: Let $g(x) = 5x^2$ and $h(x) = 5x^2 - 2x + 10$ . Then $g'(x) = 10x$ and $h'(x) = 10x - 2$ .
Simplify Numerator: Using the quotient rule, we get $f'(x) = \frac{10x(5x^2 - 2x + 10) - 5x^2(10x - 2)}{(5x^2 - 2x + 10)^2}$ .
Set Derivative Equal to Zero: Simplify the numerator of $f'(x)$ : $10x(5x^2 - 2x + 10) - 5x^2(10x - 2) = 50x^3 - 20x^2 + 100x - 50x^3 + 10x^2 = -10x^2 + 100x$ .
Factor and Solve: Now we have $f'(x) = \frac{-10x^2 + 100x}{(5x^2 - 2x + 10)^2}$ .
Identify Critical Points: The critical points occur where the derivative is zero or undefined. The denominator $(5x^2 - 2x + 10)^2$ is always positive, so the derivative is never undefined. We only need to set the numerator equal to zero to find the critical points.
Identify Critical Points: The critical points occur where the derivative is zero or undefined. The denominator $(5x^2 - 2x + 10)^2$ is always positive, so the derivative is never undefined. We only need to set the numerator equal to zero to find the critical points.Set the numerator equal to zero: $-10x^2 + 100x = 0$ .
Identify Critical Points: The critical points occur where the derivative is zero or undefined. The denominator $(5x^2 - 2x + 10)^2$ is always positive, so the derivative is never undefined. We only need to set the numerator equal to zero to find the critical points.Set the numerator equal to zero: $-10x^2 + 100x = 0$ .Factor out the common factor: $-10x(x - 10) = 0$ .
Identify Critical Points: The critical points occur where the derivative is zero or undefined. The denominator $5x^2 - 2x + 10$ ^ $2$ is always positive, so the derivative is never undefined. We only need to set the numerator equal to zero to find the critical points.Set the numerator equal to zero: (-10\)x^ $2$ + $100$ x = $0$ .Factor out the common factor: (-10\)x(x - $10$ ) = $0$ .Set each factor equal to zero: (-10\)x = $0$ and x - $10$ = $0$ .
Identify Critical Points: The critical points occur where the derivative is zero or undefined. The denominator $(5x^2 - 2x + 10)^2$ is always positive, so the derivative is never undefined. We only need to set the numerator equal to zero to find the critical points.Set the numerator equal to zero: $-10x^2 + 100x = 0$ .Factor out the common factor: $-10x(x - 10) = 0$ .Set each factor equal to zero: $-10x = 0$ and $x - 10 = 0$ .Solve for $x$ : $x = 0$ and $x = 10$ .
Identify Critical Points: The critical points occur where the derivative is zero or undefined. The denominator $5x^2 - 2x + 10$ ^ $2$ is always positive, so the derivative is never undefined. We only need to set the numerator equal to zero to find the critical points.Set the numerator equal to zero: (-10\)x^ $2$ + $100$ x = $0$ .Factor out the common factor: (-10\)x(x - $10$ ) = $0$ .Set each factor equal to zero: (-10\)x = $0$ and x - $10$ = $0$ .Solve for x: \x = $0$ and \x = $10$ .The critical points of the function are \x = $0$ and \x = $10$ .