.Let `y=f(x)=sin^(3)((pi)/(3){cos{(pi)/(3sqrt(2))(-4x^(3)+5x^(2)+1)^((3)/(2))))`.Then,at `x=1`.

Options

A. `2y^(')+sqrt(3)pi^(2)y=0`
B. `sqrt(2)y^(')-3 pi^(2)y=0`
C. `2y^\(')+3 pi^(2)y=0`
D. `y' + 3pi^2y = 0`

Answer

To solve the given problem, we need to find the value of the function \( y = f(x) = \sin^3\left(\frac{\pi}{3} \cos\left(\frac{\pi}{3\sqrt{2}} \left(-4x^3 + 5x^2 + 1\right)^{\frac{3}{2}}\right)\right) \) at \( x = 1 \).

### Step 1: Substitute \( x = 1 \) into the function
We start by substituting \( x = 1 \) into the function:

\[ y = \sin^3\left(\frac{\pi}{3} \cos\left(\frac{\pi}{3\sqrt{2}} \left(-4(1)^3 + 5(1)^2 + 1\right)^{\frac{3}{2}}\right)\right) \]

Calculating the inner expression:

\[ -4(1)^3 + 5(1)^2 + 1 = -4 + 5 + 1 = 2 \]

Now we can substitute this back into the function:

\[ y = \sin^3\left(\frac{\pi}{3} \cos\left(\frac{\pi}{3\sqrt{2}} (2)^{\frac{3}{2}}\right)\right) \]

### Step 2: Simplify the expression
Next, we simplify \( (2)^{\frac{3}{2}} \):

\[ (2)^{\frac{3}{2}} = 2\sqrt{2} \]

Now substitute this into the cosine function:

\[ y = \sin^3\left(\frac{\pi}{3} \cos\left(\frac{\pi}{3\sqrt{2}} \cdot 2\sqrt{2}\right)\right) \]

### Step 3: Calculate the argument of cosine
Now we calculate \( \frac{\pi}{3\sqrt{2}} \cdot 2\sqrt{2} \):

\[ \frac{\pi}{3\sqrt{2}} \cdot 2\sqrt{2} = \frac{2\pi}{3} \]

Thus, we have:

\[ y = \sin^3\left(\frac{\pi}{3} \cos\left(\frac{2\pi}{3}\right)\right) \]

### Step 4: Calculate \( \cos\left(\frac{2\pi}{3}\right) \)
The cosine of \( \frac{2\pi}{3} \) is:

\[ \cos\left(\frac{2\pi}{3}\right) = -\frac{1}{2} \]

### Step 5: Substitute back into the sine function
Now substitute this value back into the sine function:

\[ y = \sin^3\left(\frac{\pi}{3} \cdot -\frac{1}{2}\right) \]

This simplifies to:

\[ y = \sin^3\left(-\frac{\pi}{6}\right) \]

### Step 6: Calculate \( \sin\left(-\frac{\pi}{6}\right) \)
The sine of \( -\frac{\pi}{6} \) is:

\[ \sin\left(-\frac{\pi}{6}\right) = -\frac{1}{2} \]

### Step 7: Final calculation
Now we calculate:

\[ y = \left(-\frac{1}{2}\right)^3 = -\frac{1}{8} \]

Thus, the value of \( y \) at \( x = 1 \) is:

\[ \boxed{-\frac{1}{8}} \]