Final answer:
The linear approximation L(x) of the function g(x) = 3√(1 - x) at a = 0 is L(x) = 3, which is determined by evaluating the function and its derivative at x = 0. The correct option is a.
Step-by-step explanation:
Finding the Linear Approximation L(x) for g(x):
To find the linear approximation L(x) of the function g(x) = 3√(1 - x) at a = 0, we first need to determine the value of the function and its derivative at x = 0. The linear approximation of a function at a point a is given by L(x) = f(a) + f′(a)(x - a), where f′(a) represents the derivative of the function at a. For the function g(x), we can calculate:
- The function value at a = 0: g(0) = 3√(1 - 0) = 3.
- The derivative of g(x): g′(x) = √3(1 - x)^(-2/3)×(-1).
- The derivative value at a = 0: g′(0) = -√3.
Since the derivative of a constant is zero, the linear approximation becomes L(x) = g(0), because all terms involving x will be zero (since g′(0) = 0). Therefore, the linear approximation is L(x) = 3.
Considering the given options, answer (a) L(x) = 3 is the correct linear approximation of the function g(x) at a = 0.