Let `f : (0, oo) rarr R` be a continuous function such that `f(x) = int_(0)^(x) t f(t) dt`. If `f(x^(2)) = x^(4) + x^(5)`, then `sum_(r = 1)^(12) f(r^(2))`, is equal to

Options

A. 216
B. 219
C. 222
D. 225

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the given function
We are given a continuous function \( f : (0, \infty) \to \mathbb{R} \) such that:
\[ f(x) = \int_0^x t f(t) \, dt \]
and we also know that:
\[ f(x^2) = x^4 + x^5 \]

### Step 2: Substitute \( x \) with \( x^2 \)
We will replace \( x \) with \( x^2 \) in the original function:
\[ f(x^2) = \int_0^{x^2} t f(t) \, dt \]

### Step 3: Differentiate both sides
Now we differentiate both sides with respect to \( x \):
\[ \frac{d}{dx} f(x^2) = \frac{d}{dx} \left( \int_0^{x^2} t f(t) \, dt \right) \]
Using the chain rule on the left side and Leibniz's rule on the right side, we have:
\[ 2x f'(x^2) = x^2 f(x^2) \]

### Step 4: Substitute \( f(x^2) \)
Now, substitute \( f(x^2) = x^4 + x^5 \) into the equation:
\[ 2x f'(x^2) = x^2 (x^4 + x^5) \]

### Step 5: Simplify the equation
This simplifies to:
\[ 2x f'(x^2) = x^6 + x^7 \]
Dividing both sides by \( x \) (for \( x > 0 \)):
\[ 2 f'(x^2) = x^5 + x^6 \]

### Step 6: Solve for \( f'(x^2) \)
Now, we can express \( f'(x^2) \):
\[ f'(x^2) = \frac{x^5 + x^6}{2} \]

### Step 7: Integrate to find \( f(x^2) \)
To find \( f(x^2) \), we integrate \( f'(x^2) \):
\[ f(x^2) = \int \frac{x^5 + x^6}{2} \, dx = \frac{1}{2} \left( \frac{x^6}{6} + \frac{x^7}{7} \right) + C \]
This gives:
\[ f(x^2) = \frac{x^6}{12} + \frac{x^7}{14} + C \]

### Step 8: Use the condition \( f(x^2) = x^4 + x^5 \)
We know that:
\[ f(x^2) = x^4 + x^5 \]
Setting the two expressions equal:
\[ \frac{x^6}{12} + \frac{x^7}{14} + C = x^4 + x^5 \]

### Step 9: Determine the constant \( C \)
To find \( C \), we can equate coefficients. For \( x^6 \) and \( x^7 \), we can see that:
- The coefficient of \( x^6 \) gives \( \frac{1}{12} = 0 \) (no \( x^6 \) term on the right).
- The coefficient of \( x^7 \) gives \( \frac{1}{14} = 0 \) (no \( x^7 \) term on the right).
- The constant term gives \( C = 0 \).

### Step 10: Find \( f(r^2) \)
Thus, we have:
\[ f(x^2) = x^4 + x^5 \]
Now, we need to find:
\[ \sum_{r=1}^{12} f(r^2) = \sum_{r=1}^{12} (r^4 + r^5) \]
This can be split into two sums:
\[ \sum_{r=1}^{12} r^4 + \sum_{r=1}^{12} r^5 \]

### Step 11: Calculate the sums
Using the formulas for the sums of powers:
- The sum of the first \( n \) fourth powers is:
\[ \sum_{r=1}^{n} r^4 = \frac{n(n+1)(2n+1)(3n^2 + 3n - 1)}{30} \]
- The sum of the first \( n \) fifth powers is:
\[ \sum_{r=1}^{n} r^5 = \left( \frac{n(n+1)}{2} \right)^2 \]

For \( n = 12 \):
1. Calculate \( \sum_{r=1}^{12} r^4 \):
\[ \sum_{r=1}^{12} r^4 = \frac{12 \cdot 13 \cdot 25 \cdot 155}{30} = 6500 \]

2. Calculate \( \sum_{r=1}^{12} r^5 \):
\[ \sum_{r=1}^{12} r^5 = \left( \frac{12 \cdot 13}{2} \right)^2 = 650^2 = 422500 \]

### Step 12: Combine the results
Finally, we combine the results:
\[ \sum_{r=1}^{12} f(r^2) = 6500 + 422500 = 429000 \]

Thus, the final answer is:
\[ \boxed{429000} \]