Final answer:
To find the linearization of the function f(x) = sin^2(x) at x = -π/6, calculate the value of f at -π/6 and its derivative at this point, then use the formula L(x) = f(a) + f'(a)(x - a) to obtain the linear approximation.
Step-by-step explanation:
The question is asking to find the linear approximation or linearization of the function f(x) = σsin^{2}(x) at x = -π/6. The linearization of a function at a point is the best linear approximation of the function near that point. The linearization, L(x), can be expressed as L(x) = f(a) + f'(a)(x - a) where a is the point at which we are approximating and f'(a) is the derivative of the function at a.
To find the linearization L(x), we first compute the derivative f'(x), which for the function given is f'(x) = 2sin(x)cos(x) by using the chain rule and the trigonometric identity for sin(2x) = 2sin(x)cos(x).
Evaluating the derivative at x = -π/6, we obtain f'(-π/6). Then, we compute f(-π/6), the value of the function at x = -π/6. Finally, we use both of these to construct the linearization:
L(x) = f(-π/6) + f'(-π/6)(x + π/6).