find f(x) Knowing f'(x) = 2x^2 - 6x + 7,  find f(x) if f(0) = -8.  

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

It is given that f'(x) = 2x^2 - 6x + 7

To find f(x) we have to integrate f'(x)

Int[f'(x) dx] = Int[2x^2 - 6x + 7 dx]

=> (2/3)x^3 - 3x^2 + 7x + C

As f(0) = -8

(2/3)*0^3 - 3*0^2 + 7*0 + C = -8

=> C = -8

The function f(x) = (2/3)x^3 - 3x^2 + 7x - 8

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

If f'(x) = 2x^2 - 6x + 7, that means that the function f(x) is a polynomial of 3rd order:

f(x) = ax^3 + bx^2 + cx + d

If f(0) = -8, we'll replace x by 0 and we'll get:

f(0) = d

d = -8

f'(x) = 3ax^2 + 2bx + c

Comparing f'(x) with the given f'(x) we'll get:

3a = 2

a = 2/3

2b = -6

b = -3

c = 7

The function f(x) = 2x^3/3 - 3x^2 + 7x - 8.

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