Final answer:
To find the equation of the tangent line at x = 2, we first find the derivative of the function f(x) and substitute x = 2 into it to get the slope. Then, we use the point-slope form to write the equation of the tangent line at x = 2.
Step-by-step explanation:
To find the equation of the tangent line at x = 2, we first find the derivative of the function f(x). The derivative of f(x) is -4/(2sqrt(2x^2 + 4x))^3 - 5. We can substitute x = 2 into this derivative to find the slope of the tangent line at x = 2. Plugging x = 2 into the derivative gives us a slope of -4/12 - 5 = -4/12 - 60/12 = -64/12 = -16/3.
Next, we use the point-slope form of a linear equation, y - y1 = m(x - x1), where m is the slope and (x1, y1) is a point on the line. In this case, the point is (2, f(2)). Plugging in x = 2 into the original function f(x) gives us f(2) = -4/sqrt(8 + 8) - 5(2) = -4/(4sqrt(2)) - 10 = -1/sqrt(2) - 10. Therefore, the equation of the tangent line at x = 2 is y - (-1/sqrt(2) - 10) = (-16/3)(x - 2).