Question

consider the function (f(x)) whose second derivative is (f(x) = 8x + 8sin(x)). if (f(0) = 2) and (f(0) = 4), what is (f(5))?

Explanation:

Find the first derivative \(f'(x)\) by integration

$$ f'(x) = \int f''(x) \, dx = \int (8x + 8\sin(x)) \, dx = 4x^2 - 8\cos(x) + C_1 $$

Using the initial condition \(f'(0) = 4\):

$$ 4 = 4(0)^2 - 8\cos(0) + C_1 \implies 4 = -8 + C_1 \implies C_1 = 12 $$

$$ f'(x) = 4x^2 - 8\cos(x) + 12 $$

Find the function \(f(x)\) by integration

$$ f(x) = \int f'(x) \, dx = \int (4x^2 - 8\cos(x) + 12) \, dx = \frac{4}{3}x^3 - 8\sin(x) + 12x + C_2 $$

Using the initial condition \(f(0) = 2\):

$$ 2 = \frac{4}{3}(0)^3 - 8\sin(0) + 12(0) + C_2 \implies C_2 = 2 $$

$$ f(x) = \frac{4}{3}x^3 - 8\sin(x) + 12x + 2 $$

Evaluate \(f(5)\)

$$ f(5) = \frac{4}{3}(5)^3 - 8\sin(5) + 12(5) + 2 = \frac{500}{3} - 8\sin(5) + 60 + 2 = \frac{686}{3} - 8\sin(5) $$

Answer:

Consider the function \(f(x)\) whose second derivative is \(f''(x) = 8x + 8\sin(x)\). If \(f(0) = 2\) and \(f'(0) = 4\), what is \(f(5)\)? <blank>\(\frac{686}{3} - 8\sin(5)\)</blank>