If d/dx(f(x)) = 5x^4 - 4/x^5 such that f(1) = 0. Then f(2) - 2f(1/2) is equal to : Option 1: -2 Option 2: 2 Option 3: 0 Option 4: 1

Answer: 2. Solution: Integrating: f(x) = x^5 + 1/x^4 + C. Using f(1) = 0: 1 + 1 + C = 0 C = -2. So f(x) = x^5 + 1/x^4 - 2. f(2) = 32 + 1/16 - 2 = 30 + 1/16. f(1/2) = 1/32 + 16 - 2 = 14 + 1/32. f(2) - 2f(1/2) = 30 + 1/16 - 28 - 1/16 = 2.

Calculus

Integrals

Medium

If $\frac{d}{dx}(f(x)) = 5x^4 - \frac{4}{x^5}$ such that $f(1) = 0$ . Then $f(2) - 2f\left(\frac{1}{2}\right)$ is equal to :

$-2$

✅ Correct Option: 2

Integrating: $f(x) = x^5 + \frac{1}{x^4} + C$ .

Using $f(1) = 0$ : $1 + 1 + C = 0 \Rightarrow C = -2$ .

So $f(x) = x^5 + \frac{1}{x^4} - 2$ .

$f(2) = 32 + \frac{1}{16} - 2 = 30 + \frac{1}{16}$ .

$f(1/2) = \frac{1}{32} + 16 - 2 = 14 + \frac{1}{32}$ .

$f(2) - 2f(1/2) = 30 + \frac{1}{16} - 28 - \frac{1}{16} = 2$ .

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Solution

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CUET Mathematics Past Year Questions (Topic-Wise)

If $\frac{d}{dx}(f(x)) = 5x^4 - \frac{4}{x^5}$ such that $f(1) = 0$ . Then $f(2) - 2f\left(\frac{1}{2}\right)$ is equal to :