Final answer:
The linearization l(x) of the function f(x) = x^(3/4) at x=1 is l(x) = 1 + 0.7500(x - 1), which is the equation of the tangent line to the function at that point.
Step-by-step explanation:
To find the linearization l(x) of the function f(x) = x^(3/4) at x = 1, we need to compute the function's value and its derivative at that point. The linearization is the tangent line to the function at x = 1, which gives us an approximation to the function near that point.
The function value at x = 1 is f(1) = 1^(3/4) = 1. The derivative of f with respect to x is f'(x) = (3/4)x^(-1/4). Thus, the derivative at x = 1 is f'(1) = (3/4).
The equation of the tangent line (linearization) at x = 1 is l(x) = f(1) + f'(1)(x - 1), which simplifies to l(x) = 1 + (3/4)(x - 1). Rounded to four decimal places, the linear equation is l(x) = 1 + 0.7500(x - 1).