Find the linearization l(x) of the function at a. f(x) = x^3/4, a = 81

Solution:

Given, the function f(x) = x^3/4 = (∜x)³

We have to find the linearization L(x) of the function at a = 81.

Using the formula,

L(x) = f(a) + f’(a)(x - a)

Now,

f(x) = (∜x)³

f(a) = f(81) = ∜(81)³ = 3³ = 27

f’(x)= \(\frac{3}{4x^{\frac{1}{4}}}\)

Now, f’(a) = f’(81) = \(\frac{3}{4(81)^{\frac{1}{4}}}=\frac{3}{4(3)}\)

f’(a) = 1/4

Substituting the values of f(a) and f’(a), the function becomes

L(x) = 27 + (1/4)(x - 81)

Therefore, the linearization of f(x) = x^3/4 at a = 81 is L(x) = 27 + (1/4)(x - 81).

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Find the linearization l(x) of the function at a. f(x) = x^3/4, a = 81

Summary:

The linearization of the function f(x) = x^3/4 at a = 81 is L(x) = 27 + (1/4)(x - 81).