AI tutor
Welcome to Bytelearn!
Let’s check out your problem:
Find all critical points of the function (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:
Full solution
Q. Find all critical points of the function (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:
- Find Critical Points: To find the critical points of the function, we need to find the values of where the derivative of the function is or undefined.
- Calculate Derivative: First, we find the derivative of the function using the quotient rule, which states that if , then .
- Apply Quotient Rule: Let and . Then and .
- Simplify Numerator: Using the quotient rule, we get .
- Set Derivative Equal to Zero: Simplify the numerator of : .
- Factor and Solve: Now we have .
- Identify Critical Points: The critical points occur where the derivative is zero or undefined. The denominator 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 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: .
- Identify Critical Points: The critical points occur where the derivative is zero or undefined. The denominator 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: .Factor out the common factor: .
- Identify Critical Points: The critical points occur where the derivative is zero or undefined. The denominator ^ 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^ + x = .Factor out the common factor: (-10\)x(x - ) = .Set each factor equal to zero: (-10\)x = and x - = .
- Identify Critical Points: The critical points occur where the derivative is zero or undefined. The denominator 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: .Factor out the common factor: .Set each factor equal to zero: and .Solve for : and .
- Identify Critical Points: The critical points occur where the derivative is zero or undefined. The denominator ^ 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^ + x = .Factor out the common factor: (-10\)x(x - ) = .Set each factor equal to zero: (-10\)x = and x - = .Solve for x: \x = and \x = .The critical points of the function are \x = and \x = .
