Question

Find f
f ′ ( x ) = 8 x 3 1 x , x > 0 f ( 1 ) = − 3 ′ ( ) = 8 3 1 , > 0 ( 1 ) = − 3

Answer 1

To find function f(x), integrate the derivative f′(x)=8x^3 1/x, simplify, and use the initial condition f(1)=-3 to determine the constant of integration. The resulting function is f(x)=8/5 x^5 - 31/5.

Step-by-step explanation:

To find the function f(x) from the derivative f'(x) = 8x31/x, for x > 0, with the condition f(1) = -3, we need to integrate the given derivative.

Step 1: Integrate the derivative f'(x) to get f(x).

F(x)dx = ∫ 8x31/x dx

Step 2: Simplify and integrate the function.

F(x) = ∫ 8x4 dx

Step 3: Find the antiderivative.

F(x) = 8/5 x5 + C, where C is the integration constant.

Step 4: Use the initial condition f(1) = -3 to solve for C.

8/5(1)5 + C = -3 → C = -3 - 8/5

Step 5: Write the final function f(x).

f(x) = 8/5 x5 - 31/5

This function satisfies both the derivative and the initial condition given.