find function

Question

Find a function f(x) such that f′(x)=-2x2-1 and f(0)=1

 

f(x)= 

Answer 1

Integrating:

 

    f(x) = integral  (-2x^2 - 1) 

 

          =   (-2/3)x^3 - x + c

 

Since f(0)= 1 --->  (-2/3)(0)^3 - 0 + c = 1

                                 c = 1

 

f(x) = (-2/3)x^3 - x + 1

 

CHECK: f'(x) = -2x^2 - 1 and f(0) = 1