Questions: Compute the following. The second derivative of (2x^3 - x^2 + 6x - 7) evaluated at x=2. Simplify your answer.

Transcript text: Compute the following. \[ \left.\frac{d^{2}}{d x^{2}}\left(2 x^{3}-x^{2}+6 x-7\right)\right|_{x=2} \] $\left.\frac{d^{2}}{d x^{2}}\left(2 x^{3}-x^{2}+6 x-7\right)\right|_{x=2}=$ $\square$ (Simplify your answer.)

Solution

Solution Steps

Step 1: Define the Function

Given the function: \[ f(x) = 2x^3 - x^2 + 6x - 7 \]

Step 2: Compute the First Derivative

The first derivative of $ f(x) $ is: \[ f'(x) = \frac{d}{dx}(2x^3 - x^2 + 6x - 7) = 6x^2 - 2x + 6 \]

Step 3: Compute the Second Derivative

The second derivative of $ f(x) $ is: \[ f''(x) = \frac{d}{dx}(6x^2 - 2x + 6) = 12x - 2 \]

Step 4: Evaluate the Second Derivative at $ x = 2 $

Substitute $ x = 2 $ into the second derivative: \[ f''(2) = 12(2) - 2 = 24 - 2 = 22 \]

Final Answer

$\boxed{22}$

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