If f '(x) = 4x3 - 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x). - Mathematics and Statistics

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Sum

If f '(x) = 4x3 - 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x).

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Solution

f '(x) = 4x3 - 3x2 + 2x + k       ....[Given]

f(x) = ∫ f '(x) dx

= ∫ (4x3 - 3x2 + 2x + k) dx

= 4 ∫ x3 dx - 3 ∫ x2 dx + 2 ∫ x dx + k ∫ dx

`= 4 ("x"^4/4) - 3("x"^3/3) + 2("x"^2/2)  "kx" + "c"`

∴ f(x) = x4 - x3 + x2 + kx + c   ....(i)

Now, f(0) = 1      ...[Given]

∴ (0)4 - (0)3 + (0)2 + k(0) + c = 1

∴ c = 1               ....(ii)

Also, f(1) = 4

∴ `1^4 - 1^3 + 1^2 + "k"(1) + 1 = 4`

∴ 2 + k = 4

∴ k = 2                 ...(iii)

Substituting (ii) and (iii) in (i), we get

f(x) = x4 - x3 + x2 + 2x + 1

  Is there an error in this question or solution?
Chapter 1.5: Integration - Q.5

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