Find the Horizontal Tangent Line f(x)=x/( square root of 2x-1)

Problem

ƒ(x)=x/√(,2*x−1)

Solution

1. Identify the condition for a horizontal tangent line, which occurs when the derivative of the function is equal to zero: ƒ(x)′=0

2. Apply the quotient rule to find the derivative, where u=x and v=(2*x−1)(1/2)

3. Differentiate the numerator and denominator: d(x)/d(x)=1 and d(2*x−1)/d(x)=1/2*(2*x−1)(−1/2)*(2)=(2*x−1)(−1/2)

4. Substitute these into the quotient rule formula:

ƒ(x)′=((2*x−1)(1/2)*(1)−x*(2*x−1)(−1/2))/(((2*x−1)(1/2))2)

5. Simplify the numerator by factoring out (2*x−1)(−1/2)

ƒ(x)′=((2*x−1)(−1/2)*((2*x−1)−x))/(2*x−1)

6. Combine the terms to find the simplified derivative:

ƒ(x)′=(x−1)/((2*x−1)(3/2))

7. Set the derivative to zero and solve for x by setting the numerator equal to zero:

x−1=0⇒x=1

8. Find the y-coordinate by substituting x=1 back into the original function:

ƒ(1)=1/√(,2*(1)−1)=1

9. Write the equation of the horizontal tangent line, which is in the form y=c

Final Answer

y=1

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