The tangent line will touch one point of the original function.
Set the two equations equal to each other. Rewrite the square root as a fractional power.
We need a relationship of x and k since we have 2 unknown variables.
Take the derivative of f(x) and set the derivative equal to the slope of the tangent line equation, which is 1, and solve for x.
`f'(x) = 1/2(k)x^(-1/2)`
`2= k * (1/ x^(1/2))`
Square both sides.
`k^2 = 4x`
`x= k^2 /4`
Substitute the x back into the first equation.
`k(k^2 /4)^(1/2)= (k^2/4)+4`
`k(k/2)= k^2/4 +4`
`k^2/2 = k^2/4+4`
Subtract `k^2/4` on both sides.
`k^2 = 16`
If we plugged both numbers back to recheck, only `k=4` will work.
The answer is: