Final answer:
To find the equation for the tangent line to the graph of f(x) at x = 2, find the derivative of f(x), substitute x = 2 to find the slope of the tangent line, and use the point-slope form of the equation to find the equation of the line.
Step-by-step explanation:
To find the equation for the tangent line to the graph of f(x) at x = 2, we first find the derivative of f(x) using the power rule for differentiation. In this case, f'(x) = (1/3)(2x+4)^(-2/3)(2), which simplifies to f'(x) = 2/(3(2x+4)^(2/3)).
Next, we substitute x = 2 into the derivative to find the slope of the tangent line at x = 2. Therefore, the slope of the tangent line is f'(2) = 2/(3(2(2)+4)^(2/3)), which simplifies to 2/6 = 1/3.
Using the point-slope form of the equation of a line, we can write the equation of the tangent line as y - f(2) = (1/3)(x - 2), where f(2) is the value of f(x) at x = 2. To find f(2), substitute x = 2 into the original function f(x). Therefore, f(2) = (2(2)+4)^(1/3) = 6^(1/3) = 1.8171 (rounded to four decimal places).
Hence, the equation of the tangent line to the graph of f(x) at x = 2 is y - 1.8171 = (1/3)(x - 2). Simplifying, we get y = (1/3)x + 1.4842 (rounded to four decimal places).