Final answer:
The linear approximation of the function f(x) = x^5 - 4x^4 at a = -1 is obtained by evaluating the function and its derivative at that point. None of the provided options correctly represents the linear approximation, which should be 21x + 16.
Step-by-step explanation:
The question asks for the linear approximation of the function f(x) = x^5 - 4x^4 at a = -1. To find the linear approximation, also known as the tangent line approximation, we use the point-slope form of the linear equation and evaluate the function and its derivative at the point a = -1.
The function is f(x) = x^5 - 4x^4 and its derivative f'(x) = 5x^4 - 16x^3. Evaluating both the function and its derivative at x = -1 gives us f(-1) = -1 - 4(-1)^4 = -5 and f'(-1) = 5(-1)^4 - 16(-1)^3 = 5 + 16 = 21. Therefore, the linear approximation l(x) at a = -1 is:
l(x) = f(a) + f'(a)(x - a)
l(x) = -5 + 21(x - (-1))
l(x) = -5 + 21(x + 1)
l(x) = 21x + 16
None of the provided options a) -9x - 5, b) -6x + 3, c) x^5 + 4x^4, d) -5x - 4 matches the correct linear approximation.