If `y = (x-3) (x-2) (x-1) x(x+1)(x+2) (x+3),` then ` (d ^(2)y)/(dx ^(2)) at x=1` is:

Options

A. (a)`-101`
B. (b)`-48`
C. (c)56
D. (d)190

Answer

To find the second derivative of the function \( y = (x-3)(x-2)(x-1)x(x+1)(x+2)(x+3) \) at \( x = 1 \), we can follow these steps:

### Step 1: Rewrite the function
We can simplify the expression for \( y \) by grouping the factors:
\[ y = (x-3)(x+3)(x-2)(x+2)(x-1)x \]
Using the difference of squares, we can rewrite:
\[ y = (x^2 - 9)(x^2 - 4)(x^2 - 1)x \]

### Step 2: Expand the function
Now, we will expand \( y \):
1. First, expand \( (x^2 - 9)(x^2 - 4) \):
\[ (x^2 - 9)(x^2 - 4) = x^4 - 4x^2 - 9x^2 + 36 = x^4 - 13x^2 + 36 \]
2. Now, multiply this result by \( (x^2 - 1)x \):
\[ y = (x^4 - 13x^2 + 36)(x^2 - 1)x \]
Expanding this gives:
\[ y = x(x^4(x^2 - 1) - 13x^2(x^2 - 1) + 36(x^2 - 1)) \]
This results in:
\[ y = x(x^6 - x^4 - 13x^4 + 13x^2 + 36x^2 - 36) = x^7 - 14x^5 + 49x^3 - 36x \]

### Step 3: Differentiate the function
Now we differentiate \( y \) with respect to \( x \):
\[ \frac{dy}{dx} = 7x^6 - 70x^4 + 147x^2 - 36 \]

### Step 4: Find the second derivative
Next, we differentiate \( \frac{dy}{dx} \) to find \( \frac{d^2y}{dx^2} \):
\[ \frac{d^2y}{dx^2} = 42x^5 - 280x^3 + 294x \]

### Step 5: Evaluate the second derivative at \( x = 1 \)
Now we substitute \( x = 1 \) into \( \frac{d^2y}{dx^2} \):
\[ \frac{d^2y}{dx^2} \bigg|_{x=1} = 42(1)^5 - 280(1)^3 + 294(1) \]
Calculating this gives:
\[ = 42 - 280 + 294 = 56 \]

### Final Answer
Thus, the value of \( \frac{d^2y}{dx^2} \) at \( x = 1 \) is:
\[ \boxed{56} \]