Find the Horizontal Tangent Line y=x square root of x

Problem

y=x√(,x)

Solution

1. Rewrite the function using exponent notation to make differentiation easier.

y=x⋅x(1/2)

y=x(3/2)

2. Differentiate the function with respect to x using the power rule.

d(y)/d(x)=3/2*x(1/2)

3. Identify the condition for a horizontal tangent line, which occurs when the slope of the tangent line is zero.

d(y)/d(x)=0

4. Solve for x by setting the derivative equal to zero.

3/2*x(1/2)=0

x(1/2)=0

x=0

5. Substitute the xvalue back into the original function to find the corresponding ycoordinate.

y=0√(,0)

y=0

6. Determine the equation of the horizontal tangent line, which is in the form y=c

y=0

Final Answer

y=0

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