What is the second derivative of the function f(x) = 2x^4 - 4x^3 + 3?

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Question: What is the second derivative of the function f(x) = 2x^4 - 4x^3 + 3?

Options:

Correct Answer: 24x^2 - 24x

Solution:

First derivative f\'(x) = 8x^3 - 12x^2. Second derivative f\'\'(x) = 24x^2 - 24x.

What is the second derivative of the function f(x) = 2x^4 - 4x^3 + 3?

What is the second derivative of the function f(x) = 2x^4 - 4x^3 + 3?

Step 1: Start with the function f(x) = 2x^4 - 4x^3 + 3.
Step 2: Find the first derivative f'(x) by using the power rule. The power rule states that if you have x^n, the derivative is n*x^(n-1).
Step 3: Apply the power rule to each term in f(x):
- For 2x^4, the derivative is 4*2x^(4-1) = 8x^3.
- For -4x^3, the derivative is 3*(-4)x^(3-1) = -12x^2.
- For the constant 3, the derivative is 0.
Step 4: Combine the derivatives to get the first derivative: f'(x) = 8x^3 - 12x^2.
Step 5: Now, find the second derivative f''(x) by differentiating f'(x).
Step 6: Again, apply the power rule to each term in f'(x):
- For 8x^3, the derivative is 3*8x^(3-1) = 24x^2.
- For -12x^2, the derivative is 2*(-12)x^(2-1) = -24x.
Step 7: Combine the derivatives to get the second derivative: f''(x) = 24x^2 - 24x.

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