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Sets, Relations and Functions

Question : 23 of 65

Marks: +1, -0

f'(x)=3 x^{2}+2

f(2)

Solution:

Concept:

The derivative of a function determines its family up to a constant; without an initial condition, the function is not uniquely determined.

Explanation:

Given

f'(x) = 3x^2 + 2

Integrate:

f(x) = \int (3x^2 + 2) \, dx = x^3 + 2x + C

Then

f(2) = 2^3 + 2(2) + C = 8 + 4 + C = 12 + C

Since

C

is unknown,

f(2)

cannot be fixed to a specific value.

Answer:

The value depends on the constant of integration, so none of the given options is necessarily correct.

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