Final answer:
To find f(x) given f''(x) = 8x³+5, we need to integrate the given equation twice. By solving the integration equations and using the given initial conditions, we find that the function is f(x) = (2/5)x⁵+(5/2)x²-5x+0.5.
Step-by-step explanation:
To find f(x) given f''(x) = 8x³+5, we need to integrate the given equation twice. Let's start by integrating 8x³+5 with respect to x once to find f'(x). The integral of 8x³+5 is 2x⁴+5x+C, where C is the constant of integration. We know that f'(1) = 8, so we can substitute x=1 into the equation and solve for C. Plugging in x=1 and f'(1)=8, we get 2(1)⁴+5(1)+C = 8. Simplifying, we find C = -5.Now we have f'(x) = 2x⁴+5x-5. We can integrate f'(x) once more to find f(x). The integral of 2x⁴+5x-5 is (2/5)x⁵+(5/2)x²-5x+K, where K is the constant of integration. We know that f(1) = 0, so we substitute x=1 into the equation and solve for K. Plugging in x=1 and f(1)=0, we get (2/5)(1)⁵+(5/2)(1)²-5(1)+K=0. Simplifying, we find K = 0.5.
Therefore, the function f(x) = (2/5)x⁵+(5/2)x²-5x+0.5.