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We are given the function \( f(x) = -9x^{\frac{5}{3}} \), and we are tasked with finding and interpreting \( f'(27) \) (the first derivative) and \( f''(27) \) (the second derivative).

### Step 1: Find \( f'(x) \)
We differentiate \( f(x) \) with respect to \( x \):

\[
f(x) = -9x^{\frac{5}{3}}
\]
The derivative of \( x^{\frac{5}{3}} \) is given by:
\[
\frac{d}{dx}\left(x^{\frac{5}{3}}\right) = \frac{5}{3}x^{\frac{2}{3}}
\]
So, applying the chain rule to \( f(x) \):
\[
f'(x) = -9 \times \frac{5}{3}x^{\frac{2}{3}} = -15x^{\frac{2}{3}}
\]

### Step 2: Find \( f'(27) \)
Substitute \( x = 27 \) into \( f'(x) \):
\[
f'(27) = -15(27^{\frac{2}{3}})
\]
Since \( 27^{\frac{2}{3}} \) is the cube root of 27 squared:
\[
27^{\frac{2}{3}} = (3^3)^{\frac{2}{3}} = 3^2 = 9
\]
Thus:
\[
f'(27) = -15 \times 9 = -135
\]

### Step 3: Find \( f''(x) \)
Now, differentiate \( f'(x) = -15x^{\frac{2}{3}} \) with respect to \( x \):
\[
f''(x) = -15 \times \frac{2}{3}x^{-\frac{1}{3}} = -10x^{-\frac{1}{3}}
\]

### Step 4: Find \( f''(27) \)
Substitute \( x = 27 \) into \( f''(x) \):
\[
f''(27) = -10(27^{-\frac{1}{3}})
\]
Since \( 27^{-\frac{1}{3}} = \frac{1}{3} \):
\[
f''(27) = -10 \times \frac{1}{3} = -\frac{10}{3}
\]

### Interpretation
At \( x = 27 \):
- \( f'(27) = -135 \), meaning the function is decreasing at a rate of 135 units per unit increase in \( x \).
- \( f''(27) = -\frac{10}{3} \), meaning the rate of decrease of the function is slowing down, indicating concavity.

Would you like more details on this, or do you have any questions?

Here are 5 related questions you might consider:
1. What does the concavity of a function tell us about its behavior?
2. How can we determine where a function has inflection points?
3. Why is the derivative of a function important for understanding motion or rates of change?
4. How would you find the third derivative of the given function?
5. What role do fractional exponents play in derivatives and integrals?

**Tip**: When working with fractional exponents, always remember to simplify the expressions carefully to avoid errors with roots and powers.

Find and interpret f'(27) and f''(27) for the function f(x) = -9x^(5/3).

Learn how to find and interpret the first and second derivatives of the function f(x) = -9x^(5/3) at x = 27. This problem involves key calculus concepts like the power rule and chain rule, suitable for students in Grades 11-12.

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