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प्रश्न
If f'(x) = 9x2 – 6x and f(0) = – 3 find f(x)
उत्तर
Given f’(x) = 9x2 – 6x and f(0) = – 3
`("d"f(x))/("d"x)` = 9x2 – 6x
df(x) = (9x2 – 6x) dx
`int "d"f(x) = int (9x^2 - 6x)`
`int "d"f(x) = int 9x^2 "d"x - int 6x "d"x`
f(x) = `9 int x^2 "d"x - 6 int x "d"x`
f(x) = `9 xx x^3/3 - 6 x^2/2 + "c"`
f(x) = 3x3 – 3x2 + c .........(1)
f(0) = – 3
(1) ⇒ f(0) = 3 × – 3 . 02 + c
– 3 = 0 – 0
⇒ c = – 3
Substituting in equation (1) we get
f(x) = 3x3 – 3x2 – 3
f(x) = 3(x3 – x2 – 1)
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