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.
kx^(1/2)= x+4
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)= kx^(1/2)
f'(x) = 1/2(k)x^(-1/2)
1=1/2(k)x^(-1/2)
2= kx^(-1/2)
2= k * (1/ x^(1/2))
k=2x^(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^2/4=4
k^2 = 16
k=+-4
If we plugged both numbers back to recheck, only k=4 will work.
The answer is:
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