Mathematics

Mathematics

Problem

22. If ddxf(x)=4x3−3x4\frac{d}{d x} f(x)=4 x^{3}-\frac{3}{x^{4}} such that f(2)=0f(2)=0. Then f(x)f(x) is (A) x4+1x3−1298x^{4}+\frac{1}{x^{3}}-\frac{129}{8} (B) x3+1x4+1298x^{3}+\frac{1}{x^{4}}+\frac{129}{8} (C) x4+1x3+1298x^{4}+\frac{1}{x^{3}}+\frac{129}{8} (D) x3+1x4−1298x^{3}+\frac{1}{x^{4}}-\frac{129}{8} (A) x4+1x3−1298x^{4}+\frac{1}{x^{3}}-\frac{129}{8} (B) x3+1x4+1298x^{3}+\frac{1}{x^{4}}+\frac{129}{8} (C) x4+1x3+1298x^{4}+\frac{1}{x^{3}}+\frac{129}{8} (D) x3+1x4−1298x^{3}+\frac{1}{x^{4}}-\frac{129}{8}
Intuitive explanation

You're reversing differentiation here—integrating the derivative to recover the original function. Find the antiderivative of each term using the power rule, then use the initial condition f(2)=0f(2) = 0 to pin down the constant of integration. The key is treating −3x4-\frac{3}{x^4} as −3x−4-3x^{-4} so the power rule applies smoothly.

Intuitive explanation

You're reversing differentiation here—integrating the derivative to recover the original function. Find the antiderivative of each term using the power rule, then use the initial condition f(2)=0f(2) = 0 to pin down the constant of integration. The key is treating −3x4-\frac{3}{x^4} as −3x−4-3x^{-4} so the power rule applies smoothly.

Browse the problem library