If $$\frac{d}{dx}(f(x)) = 4x^{3} - \frac{3}{x^{4}}$$​ and $$f(2) = 0,$$  then  $$f(x)$$ is:

Correct option is B

Given:
​$$\frac{df}{dx} = 4x^{3} - \frac{3}{x^{4}}$$​​
Concept used:
Integrate both sides to find $$f(x).$$​​
Solution:
​$$f(x) = \int \left(4x^{3} - \frac{3}{x^{4}}\right) dx \\= \int 4x^{3} dx - \int 3x^{-4} dx \\= x^{4} + x^{-3} + C \\= x^{4} + \frac{1}{x^{3}} + C$$​​
Given $$f(2) = 0:$$​​
​$$0 = 2^{4} + \frac{1}{2^{3}} + C \\ \ \\0 = 16 + \frac{1}{8} + C \\ \ \\C = -\frac{129}{8}$$​​
Thus, $$f(x) = x^{4} + \frac{1}{x^{3}} - \frac{129}{8}$$​​
Correct answer is (b) $$\ x^{4} + \frac{1}{x^{3}} - \frac{129}{8}$$​​