f(x)= third root of radical x, a=125. Find the linearization L(x ) of the function at a. L(x)? 

To find the linearization \( L(x) \) of the function \( f(x) = \sqrt[3]{\sqrt{x}} \) at \( a = 125 \), we use the following formula for linearization:

\[
L(x) = f(a) + f'(a)(x - a)
\]

### Step 1: Find \( f(a) \)
First, we evaluate the function at \( a = 125 \).

\[
f(125) = \sqrt[3]{\sqrt{125}} = \sqrt[3]{\sqrt{125}} = \sqrt[3]{11.18} \approx 2.234
\]

### Step 2: Find \( f'(x) \)
Now we need to compute the derivative \( f'(x) \). The function \( f(x) = \sqrt[3]{x^{1/2}} \) can be written as:

\[
f(x) = x^{1/6}
\]

Using the power rule to differentiate:

\[
f'(x) = \frac{1}{6} x^{-5/6}
\]

### Step 3: Evaluate \( f'(a) \)
Now, evaluate the derivative at \( a = 125 \):

\[
f'(125) = \frac{1}{6} \cdot 125^{-5/6}
\]

Since \( 125 = 5^3 \), we can simplify \( 125^{-5/6} \):

\[
125^{-5/6} = (5^3)^{-5/6} = 5^{-5/2} = \frac{1}{5^{5/2}} = \frac{1}{55.9} \approx 0.01788
\]

Thus:

\[
f'(125) = \frac{1}{6} \times 0.01788 \approx 0.00298
\]

### Step 4: Write the linearization \( L(x) \)
Now we can plug the values into the linearization formula:

\[
L(x) = f(125) + f'(125)(x - 125)
\]

\[
L(x) \approx 2.234 + 0.00298(x - 125)
\]

So the linearization is approximately:

\[
L(x) = 2.234 + 0.00298(x - 125)
\]

This is the linear approximation of \( f(x) \) at \( a = 125 \).

---

Would you like more details on any of the steps?

Here are 5 related questions to explore:
1. How does the linearization approximation change for different values of \( a \)?
2. What are some practical applications of linearization in real-world problems?
3. How do you find the second derivative of \( f(x) = x^{1/6} \)?
4. How accurate is the linear approximation near \( a = 125 \)?
5. How can we improve the approximation using higher-order terms?

**Tip:** Linearization provides a quick way to approximate function values near a point without needing to compute the full function.

f(x)= third root of radical x, a=125. Find the linearization L(x ) of the function at a. L(x)?

Find the linearization L(x) of the function f(x) = ∛√x at a = 125. This problem explores the concepts of linearization, derivatives, and power functions, providing a detailed step-by-step solution. Suitable for high school students in Grades 11-12.

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