If d/d x {f}(x)=4 x³-{3}{x^{4}} such that {f}(2)=0, then {f}(x) is equal to

Correct Answer is : x4+1x3−1298x^{4}+{1}{x³}-129/8x4+x31−8129

Correct Answer is : x4+1x3−1298x^{4}+{1}{x³}-129/8x4+x31−8129

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MHTCET 10 May 2024 Shift 2 Paper

Question : 137 of 150

Marks: +1, -0

\frac{d}{d x} \mathrm{f}(x)=4 x^{3}-\frac{3}{x^{4}}

\mathrm{f}(2)=0

\mathrm{f}(x)

is equal to

Solution:

Concept:

Integrate the given derivative, then use the initial condition

\mathrm{f}(2)=0

to find the constant of integration.

Explanation:

Given:

\frac{d}{dx}\mathrm{f}(x)=4x^3-\frac{3}{x^4}

Rewrite the expression as

4x^3-3x^{-4}

Integrate both sides with respect to

x

\mathrm{f}(x)=\int\left(4x^3-3x^{-4}\right)\,dx

\therefore \mathrm{f}(x)=x^4+\frac{1}{x^3}+C

Use the condition

\mathrm{f}(2)=0

2^4+\frac{1}{2^3}+C=0

16+\frac{1}{8}+C=0

\therefore C=-\frac{129}{8}

Substitute

C

back:

\mathrm{f}(x)=x^4+\frac{1}{x^3}-\frac{129}{8}

Answer:

Option B,

x^{4}+\frac{1}{x^{3}}-\frac{129}{8}

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