The line y=2x + b is tangent to the graph y= sqrt x at the point P= (a,sqrt a). Find P and determine b

Answer 1

Recall that sqrt(x) = x^(1/2), so the slope of the graph at any point x is 1/(2*sqrt(x)).

So, at point P, x=a, slope is 1/(2*sqrt(a)).

So, the line is y=2x + b, meaning that the slope is 2. That means that 2 = 1/(2*sqrt(a)). So, a=1/16.

P = (1/16, 1/4)

y=2x+b
1/4 = 2(1/16)+b
1/4 = 1/8 + b
1/8 = b

y = 2x + 1/8