If $f(x)$ and $g(x)$ are two functions from R to R such that $(fog)(x)=(x^{3}-x^{2}+2)^{8},$ then $f^{\prime}(1)g^{\prime}(1)$ is :

Question

Accepted Answer

To solve the problem and justify why Option 1 is correct, we need to analyze the given composition of functions and their derivatives.

We are given that $(f \circ g)(x) = (x^3 - x^2 + 2)^8$. This means that $f(g(x)) = (x^3 - x^2 + 2)^8$.

To find $f'(1)g'(1)$, we need to use the chain rule for differentiation. The chain rule states that if $h(x) = f(g(x))$, then the derivative $h'(x)$ is given by:

$$ h'(x) = f'(g(x)) \cdot g'(x) $$

First, differentiate the right-hand side of the equation:

$$ \frac{d}{dx}[(x^3 - x^2 + 2)^8] = 8(x^3 - x^2 + 2)^7 \cdot (3x^2 - 2x) $$

Now, evaluate this derivative at $x = 1$:

1. Calculate $x^3 - x^2 + 2$ at $x = 1$:
   $$ 1^3 - 1^2 + 2 = 1 - 1 + 2 = 2 $$

2. Substitute into the derivative:
   $$ 8(2)^7 \cdot (3(1)^2 - 2(1)) = 8 \cdot 128 \cdot (3 - 2) = 8 \cdot 128 \cdot 1 = 1024 $$

Now, apply the chain rule at $x = 1$:

$$ f'(g(1)) \cdot g'(1) = 1024 $$

Since $g(1) = 2$ (from the calculation above), we have:

$$ f'(2) \cdot g'(1) = 1024 $$

To find $f'(2)$, consider that $f(g(x)) = (x^3 - x^2 + 2)^8$ implies $f(y) = y^8$ when $y = g(x)$. Therefore, $f'(y) = 8y^7$.

Evaluate $f'(2)$:

$$ f'(2) = 8 \cdot 2^7 = 8 \cdot 128 = 1024 $$

Thus, we have:

$$ 1024 \cdot g'(1) = 1024 $$

Solving for $g'(1)$, we find:

$$ g'(1) = \frac{1024}{1024} = 1 $$

Therefore, $f'(1)g'(1) = 8 \cdot 1 = 8$.

This confirms that the correct answer is Option 1: 8.

**Explanation for Incorrect Options:**

- **Option 2 (16):** This would imply that $f'(1)g'(1) = 16$, which contradicts our calculation showing $f'(1)g'(1) = 8$.
- **Option 3 (12):** Similarly, this value does not match the derived product of the derivatives, which is 8.
- **Option 4 (24):** This is also incorrect as it does not align with the calculated result of 8.

Thus, Option 1 is the only correct answer based on the given problem and calculations.