The domain of the funciton `f(x)` given by `3^(x) + 3^(f) = "min" (2t^(3) - 15t^(2) + 36+ - 25, 2 + |sin t| , 2 le t le 4)` is

Options

A. `(-oo, 1 )`
B. `(-oo, log_3 e )`
C. `(0, log_3 2 )`
D. `(-oo, log_3 2 )`

Explanation

To find the domain of the function \( f(x) \) defined by the equation

\[ 3^x + 3^f = \min(2t^3 - 15t^2 + 36t - 25, 2 + |\sin t|) \quad \text{for } 2 \leq t \leq 4, \]

we need to follow these steps:

### Step 1: Define the Functions
Let:
- \( g(t) = 2t^3 - 15t^2 + 36t - 25 \)
- \( p(t) = 2 + |\sin t| \)

We need to find the minimum of these two functions over the interval \( [2, 4] \).

### Step 2: Find the Derivative of \( g(t) \)
To find the critical points of \( g(t) \), we first compute its derivative:

\[ g'(t) = 6t^2 - 30t + 36 \]

### Step 3: Set the Derivative to Zero
Setting \( g'(t) = 0 \):

\[ 6t^2 - 30t + 36 = 0 \implies t^2 - 5t + 6 = 0 \]

Factoring gives:

\[ (t - 3)(t - 2) = 0 \]

Thus, the critical points are \( t = 2 \) and \( t = 3 \).

### Step 4: Determine the Nature of Critical Points
To find out if these points are minima or maxima, we compute the second derivative:

\[ g''(t) = 12t - 30 \]

Now, evaluate \( g''(t) \) at the critical points:

- For \( t = 2 \):
\[ g''(2) = 12(2) - 30 = 24 - 30 = -6 \quad (\text{local maximum}) \]

- For \( t = 3 \):
\[ g''(3) = 12(3) - 30 = 36 - 30 = 6 \quad (\text{local minimum}) \]

### Step 5: Evaluate \( g(t) \) at Critical Points and Endpoints
Next, we evaluate \( g(t) \) at the critical points and the endpoints of the interval \( [2, 4] \):

- \( g(2) = 2(2^3) - 15(2^2) + 36(2) - 25 = 16 - 60 + 72 - 25 = 3 \)
- \( g(3) = 2(3^3) - 15(3^2) + 36(3) - 25 = 54 - 135 + 108 - 25 = 2 \)
- \( g(4) = 2(4^3) - 15(4^2) + 36(4) - 25 = 128 - 240 + 144 - 25 = 7 \)

### Step 6: Find the Minimum Value
From the evaluations:
- \( g(2) = 3 \)
- \( g(3) = 2 \)
- \( g(4) = 7 \)

The minimum value of \( g(t) \) on the interval \( [2, 4] \) is \( 2 \) at \( t = 3 \).

### Step 7: Analyze \( p(t) \)
Now, we analyze \( p(t) = 2 + |\sin t| \):
- The minimum value of \( |\sin t| \) is \( 0 \) and maximum is \( 1 \).
- Thus, \( p(t) \) ranges from \( 2 \) to \( 3 \).

### Step 8: Find the Overall Minimum
The overall minimum value between \( g(t) \) and \( p(t) \) is:

\[ \min(2, 2) = 2 \]

### Step 9: Solve for \( f \)
We have:

\[ 3^x + 3^f = 2 \]

Rearranging gives:

\[ 3^f = 2 - 3^x \]

### Step 10: Find the Domain of \( f \)
For \( 3^f \) to be defined, we need:

\[ 2 - 3^x > 0 \implies 2 > 3^x \implies x < \log_3 2 \]

### Conclusion
Thus, the domain of the function \( f(x) \) is:

\[ (-\infty, \log_3 2) \]