Final answer:
To find the equation of the tangent line to f(x) = (x− 2)(4 − x)³ at x = 3, we need to find the derivative of the function and substitute x = 3. The slope of the tangent line is 2, so the equation of the tangent line is y = 2x - 3.
Step-by-step explanation:
To determine the equation of the tangent line to the function f(x) = (x− 2)(4 − x)³ at x = 3, we need to find the derivative of the function first. Taking the derivative of f(x) gives us f'(x) = -3(x-4)²(x-2) + (4-x)³.
Next, we substitute x = 3 into the derivative to find the slope of the tangent line at x = 3. f'(3) = -3(3-4)²(3-2) + (4-3)³ = -3(-1)²(1) + (1)³ = 3 - 1 = 2.
Finally, we use the point-slope form of a line to find the equation of the tangent line. The point-slope form is y - y₁ = m(x - x₁), where (x₁, y₁) is the point of tangency and m is the slope. So, the equation of the tangent line is y - f(3) = 2(x - 3). Simplifying this equation gives us the final result, y = 2x - 3.