Find all values of x (if any) where the tangent line to the graph of the given e

Question

Find all values of x (if any) where the tangent line to the graph of the given equation is horizontal. HINT [The tangent line is horizontal when its slope is zero.] (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)

Find all values of x (if any) where the tangent line to the graph of the given equation is horizontal. HINT [The tangent line is horizontal when its slope is zero.] (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.) f(x)=6x-6 sqrt(x) x=??

Explanation / Answer

f(x) = 6x - 6sqrt(x)

f(x) = 6x - 6(x)^(1/2)

Derive to find the slope of tangent line :

f'(x) = 6 - 6(1/2)x^(1/2 - 1)

f'(x) = 6 - 3x^(-1/2)

f'(x) = 6 - 3/sqrt(x)

Now, slope must be 0 if the tangent line is to be horizontal....

6 - 3/sqrt(x) = 0

6 = 3/sqrt(x)

sqrt(x) = 3/6

sqrt(x) = 1/2

Squaring :

x = (1/2)^2

x = 1/4 -----> ANSWER

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