Mark the correct alternative in each of the following:

If $f (x) = 1 - x + x^{2} - x^{3} + . . . - x^{99} + x^{100}$ , then $f' (1)$ equals

(a) 150 (b) −50 (c) −150 (d) 50

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Solution

$f (x) = 1 - x + x^{2} - x^{3} + . . . - x^{99} + x^{100}$

Differentiating both sides with respect to x, we get

$f' (x) = \frac{d}{d x} (1 - x + x^{2} - x^{3} + . . . - x^{99} + x^{100}) = \frac{d}{d x} (1) - \frac{d}{d x} (x) + \frac{d}{d x} (x^{2}) - \frac{d}{d x} (x^{3}) + . . . - \frac{d}{d x} (x^{99}) + \frac{d}{d x} (x^{100}) = 0 - 1 + 2 x - 3 x^{2} + . . . - 99 x^{98} + 100 x^{99} = - 1 + 2 x - 3 x^{2} + . . . - 99 x^{98} + 100 x^{99}$

Putting x = 1, we get

$f' (1) = - 1 + 2 - 3 + . . . - 99 + 100 = (- 1 + 2) + (- 3 + 4) + (- 5 + 6) + . . . + (- 99 + 100) = 1 + 1 + 1 + . . . + 1 (50 terms) = 50$

Hence, the correct answer is option (d).

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