We are given the function:

$f(x) = m \sin(Bx) + 15$ with $m > 0$, and the minimum value of the function is $-15$.

Step-by-step solution:

1. Minimum Value of the Function: The sine function, $\sin(Bx)$, varies between -1 and 1. Therefore, the function $f(x) = m \sin(Bx) + 15$ will have its minimum value when $\sin(Bx) = -1$.

   • When $\sin(Bx) = -1$, we get: $f(x) = m(-1) + 15 = -m + 15$ We are told that the minimum value is $-15$, so: $-m + 15 = -15$ Solving for $m$: $-m = -15 - 15$ $m = 30$

2. Find the Value of $f\left( \frac{\pi}{6B} \right)$: We now need to find the value of the function at $x = \frac{\pi}{6B}$. First, evaluate $\sin\left( B \cdot \frac{\pi}{6B} \right)$. This simplifies to: $\sin\left( \frac{\pi}{6} \right) = \frac{1}{2}$ Therefore, the function at $x = \frac{\pi}{6B}$ becomes: $f\left( \frac{\pi}{6B} \right) = m \sin\left( \frac{\pi}{6B} \right) + 15 = 30 \cdot \frac{1}{2} + 15$ $f\left( \frac{\pi}{6B} \right) = 15 + 15 = 30$

Thus, the value of $f\left( \frac{\pi}{6B} \right)$ is 30.

Final Answer:

The correct option is 30.

Do you want further details on any step or another explanation?

Here are some related questions for practice:

1. How would the function change if the maximum value were given instead of the minimum?
2. What is the amplitude of the function $f(x) = m \sin(Bx) + 15$?
3. How does changing the value of $B$ affect the period of the function?
4. What happens to the minimum value if $m$ is doubled?
5. Can you find the value of $f(x)$ at $x = 0$?

Tip: When analyzing sine or cosine functions, always check their amplitude and phase shifts to understand how the graph behaves.