Question

​​​​​​​Find f,
f'(x)=√(x)(9+5 x), f(1)=10

Answer 1

To find the function f(x) given f'(x), we need to integrate f'(x) with respect to x. The function f(x) = 2/3(3x√(x^2+9)+x^3+1) satisfies the given conditions f'(x) = √(x)(9+5x) and f(1) = 10.

Step-by-step explanation:

To find the function f(x) given f'(x), we need to integrate f'(x) with respect to x. In this case, f'(x) = √(x)(9+5x). Integrating this expression gives us f(x) = 2/3(3x√(x^2+9)+x^3+C), where C is the constant of integration. To find the value of C, we can use the given initial condition f(1) = 10. Substituting x = 1 and f(x) = 10 into the equation for f(x) will allow us to solve for C:

10 = 2/3(3(1)√(1^2+9)+1^3+C) = 2(3+1+C)

10 = 2(4+C)

10 = 8+2C

2C = 10-8

2C = 2

C = 1

So the function f(x) = 2/3(3x√(x^2+9)+x^3+1).