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Find the linearization $L(x)$ of the function $f(x)=x^{3 / 4}$ at $x=1$.

Answer: $L(x)=$

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Answer

$\boxed{L(x) = 1 + \frac{3}{4}(x - 1)}$ is the linearization of the function $f(x) = x^{3/4}$ at $x = 1$.

Steps

Step 1 ：Find $f(a)$ where $a = 1$, so $f(a) = f(1) = (1)^{3/4} = 1$.

Step 2 ：Find $f'(a)$. The derivative of $f(x) = x^{3/4}$ is $f'(x) = \frac{3}{4}x^{-1/4}$. So, $f'(1) = \frac{3}{4}(1)^{-1/4} = \frac{3}{4}$.

Step 3 ：Substitute $a$, $f(a)$, and $f'(a)$ into the formula to get the linearization of the function: $L(x) = 1 + \frac{3}{4}(x - 1)$.

Step 4 ：$\boxed{L(x) = 1 + \frac{3}{4}(x - 1)}$ is the linearization of the function $f(x) = x^{3/4}$ at $x = 1$.