Question

Find the linearization L(x) of the function at a. f(x) = x3/4, a = 81

Answer 1

Please note that your x^3/4 is ambiguous.  Did you mean (x^3)  divided by 4
or did you mean x to the power (3/4)?  I will assume you meant the first, not the second.  Please use the "^" symbol to denote exponentiation.

If we have a function f(x) and its derivative f'(x), and a particular x value (c) at which to begin, then the linearization of the function f(x) is

f(x) approx. equal to  [f '(c)]x + f(c)].

Here a = c = 81.

Thus, the linearization of the given function at a = c = 81 is

f(x) (approx. equal to)  3(81^2)/4 + [81^3]/4

Note that f '(c) is the slope of the line and is equal to (3/4)(81^2), and f(c) is the function value at x=c, or (81^3)/4.

What is the linearization of f(x) = (x^3)/4, if c = a = 81?

It will be f(x) (approx. equal to)