https://file.aiquickdraw.com/mathbot/file/1724561236171xbe13nyt.jpg

Let's analyze the problem shown in the image.

We are given the derivative of a function \( f'(x) = 6\left(\frac{2}{x^2} - \sqrt{x}\right) \), and the question asks whether the function \( f(x) = -\frac{12}{x} - 2\sqrt{x} \) is correct.

### Step 1: Integrate \( f'(x) \) to find \( f(x) \)

Given:
\[
f'(x) = 6\left(\frac{2}{x^2} - \sqrt{x}\right)
\]

We can split this into:
\[
f'(x) = 6 \cdot \frac{2}{x^2} - 6 \cdot \sqrt{x}
\]
\[
f'(x) = \frac{12}{x^2} - 6x^{\frac{1}{2}}
\]

Now, integrate each term separately.

1. Integrate \(\frac{12}{x^2}\):
   \[
   \int \frac{12}{x^2} \, dx = \int 12x^{-2} \, dx = 12 \cdot \left(-\frac{1}{x}\right) = -\frac{12}{x}
   \]

2. Integrate \(-6x^{\frac{1}{2}}\):
   \[
   \int -6x^{\frac{1}{2}} \, dx = -6 \cdot \frac{2}{3} x^{\frac{3}{2}} = -4x^{\frac{3}{2}} = -4\sqrt{x} \cdot x
   \]

So, after integrating, we have:
\[
f(x) = -\frac{12}{x} - 4x^{\frac{3}{2}} + C
\]

### Step 2: Compare with the given function

The function given is:
\[
f(x) = -\frac{12}{x} - 2\sqrt{x}
\]

This does **not** match the function we obtained by integrating \( f'(x) \), which had a term \(-4\sqrt{x}\) instead of \(-2\sqrt{x}\).

### Conclusion:

The function \( f(x) = -\frac{12}{x} - 2\sqrt{x} \) is **not** correct, hence the statement is **False**.

Would you like more details or have any questions?

Here are some related questions to deepen your understanding:

1. What happens if we integrate a different derivative function?
2. How does the constant of integration affect the general solution?
3. Can the function be verified by differentiating again?
4. How do different methods of integration apply to more complex functions?
5. What are the common mistakes to avoid in integrating power functions?

**Tip:** Always remember to include the constant of integration when solving indefinite integrals.

Explore the analysis of derivatives and integrals to verify the correctness of a function. This problem involves differentiating and integrating functions, applying the Fundamental Theorem of Calculus and integration techniques for power functions. Suitable for advanced high school students and beyond.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation