Final answer:
To find the linearization of the function f(x) = x^(3/4) at x = 1, we first find the derivative of the function. The derivative of f(x) is f'(x) = (3/4)x^(-1/4). Next, we evaluate f'(x) at x = 1 to find the slope of the tangent line. Finally, using the point-slope form of a line, we determine the linearization of f(x) at x = 1.
Step-by-step explanation:
To find the linearization of the function f(x) = x^(3/4) at x = 1, we first find the derivative of the function. The derivative of f(x) is f'(x) = (3/4)x^(-1/4). Next, we evaluate f'(x) at x = 1 to find the slope of the tangent line. Plugging in x = 1 into f'(x) gives us f'(1) = (3/4)(1)^(-1/4) = 3/4.
Now, we need the point (1, f(1)) to determine the equation of the tangent line. Plugging in x = 1 into f(x) gives us f(1) = 1^(3/4) = 1.
Using the point-slope form of a line, the linearization of f(x) = x^(3/4) at x = 1 is given by l(x) = f(1) + f'(1)(x - 1). Substituting in the values we found, we get l(x) = 1 + (3/4)(x - 1).