Answer:
Sure, let's solve this problem step by step.
Part 1: Find the anti-derivative of f''(x) to get f'(x).
To find the anti-derivative of f''(x), you need to integrate it with respect to x:
∫f''(x) dx = ∫(8x^2 - 8x^3 + 10x + 7) dx
Now, let's find the anti-derivative term by term:
∫8x^2 dx = (8/3)x^3 + C₁, where C₁ is the constant of integration for this term.
∫-8x^3 dx = (-8/4)x^4 + C₁ = -2x^4 + C₁
∫10x dx = 5x^2 + C₁
∫7 dx = 7x + C₁
Now, sum up all these anti-derivative terms:
f'(x) = (8/3)x^3 - 2x^4 + 5x^2 + 7x + C₁
Part 2: Find the anti-derivative for f'(x) to get f(x).
To find the anti-derivative of f'(x), you need to integrate it with respect to x:
∫f'(x) dx = ∫[(8/3)x^3 - 2x^4 + 5x^2 + 7x + C₁] dx
Now, let's find the anti-derivative term by term:
∫(8/3)x^3 dx = (8/12)x^4 + C₂ = (2/3)x^4 + C₂, where C₂ is the constant of integration for this term.
∫(-2x^4) dx = (-2/5)x^5 + C₂
∫5x^2 dx = (5/3)x^3 + C₂
∫7x dx = (7/2)x^2 + C₂
∫C₁ dx = C₁x + C₂
Now, sum up all these anti-derivative terms:
f(x) = (2/3)x^4 - (2/5)x^5 + (5/3)x^3 + (7/2)x^2 + C₁x + C₂
You now have the anti-derivative of f'(x) with the constants of integration C₁ and C₂. To determine the values of C₁ and C₂, you can use the initial conditions given:
f'(1) = -8 and f(1) = -4
Plug in x = 1 into the expressions for f'(x) and f(x), and set them equal to the given values:
f'(1) = (8/3)(1)^3 - 2(1)^4 + 5(1)^2 + 7(1) + C₁ = -8
f(1) = (2/3)(1)^4 - (2/5)(1)^5 + (5/3)(1)^3 + (7/2)(1)^2 + C₁(1) + C₂ = -4
Now, you can solve this system of equations for C₁ and C₂.
Explanation: