Question

Which of the following is not true

Easy

Solution

$\log_{e} (1 + x) - x = \log_{e} (1 + x) - \log_{e} e^{x} = \log_{e} \frac{1 + x}{e^{x}}$
$= \ln \frac{1 + x}{1 + x + \frac{x^{2}}{2!} + \frac{x^{3}}{3!} + . . .} < 0, as 1 + x < 1 + x + \frac{x^{2}}{2!} + . . . + ∴ \log_{e} (1 + x) < x, for x > 0 \frac{x}{1 + x} - \log (1 + x) = 1 - \frac{1}{1 + x} - \log (1 + x) = 1 - [\frac{1}{1 + x} + \log (1 + x)] < 0, for n > 0 ∴ \frac{x}{1 + x} < \log (1 + x), ∴ (b) is true e^{x} - (1 + x) = 1 + x + \frac{x^{2}}{2!} + \frac{x^{3}}{3!} + . . . - (1 + x) = \frac{x^{2}}{2!} + \frac{x^{3}}{3!} + . . . > 0, for x > 0 ∴ e^{x} > 1 + x, for x > 0; ∴ (c) is true e^{x} - (1 - x) = 1 + x + \frac{x^{2}}{2!} + . . . - 1 + x = 2 x + \frac{x^{2}}{2!} + \frac{x^{3}}{3!} + . . . > 0, for x > 0 ∴ e^{x} > 1 - x, for x > 0 Thus, e^{x} < (1 - x), for x > 0 is not true .$

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