for what value of x is the line tangent to y=x^2 parallel to the line tangent to y= √ x

Answer 1

The slope of the tangent line of y = x² is the derivative function. The derivative of y = x² is y' = 2x.

The slope of the tangent line of y = √x is the derivative function. The derivative of y =√x is y' = 1(2√x).

If the lines are parallel, then their slopes are equal so 2x must equal 1(2√x).

2x = 1(2√x)

2x(2√x).= 1

4x^3/2 = 1

x^3/2 = 1/4

x = (1/4)^2/3

x = 1/cuberoot(16) = 1/(2cuberoot(2)) = cuberoot(4)/4 ≈ 0.39685

Answer 2

f(x) = x^2 and g(x) = sqrt(x) = x^(1/2)

f ' (x) = 2x and g ' (x) = (1/2) x ^ (-1/2)

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

2x *x^(1/2) = (1/2)

2 x ^ (3/2) = 1/2

x^ (3/2) = 1/4

(3/2) log x = log ( 1/4)

log x = (2/3) log (1/4)

x = 10^ [(2/3)log (1/4)]

which is APPROXIMATELY 0.396850263-