First slide

Maxima and minima

Question

The maximum value of the function of $f (x) = \frac{(1 + x)^{0 .6}}{1 + x^{0 .6}}$ in the interval [0, 1] is

Easy

A

2^0.4
B

2^-0.4
C

1
D

2^0.6

Solution

$f^{'} (x) = \frac{0 .6 (1 + x)^{- 0 .4} (1 + x^{0 .6}) - 0 .6 x^{- 0 .4} (1 + x)^{0 .6}}{{(1 + x^{0 .6})}^{2}} = 0 .6 \frac{(1 + x^{0 .6}) - x^{- 0 .4} (1 + x)^{1}}{{(1 + x^{0 .6})}^{2} (1 + x)^{0 .4}} = 0 .6 \frac{(1 + x^{0 .6}) x^{0 .4} - (1 + x)}{{(1 + x^{0 .6})}^{2} (1 + x)^{0 .4} x^{0 .4}} = 0 .6 \frac{x^{0 .4} - 1}{{(1 + x^{0 .6})}^{2} (1 + x)^{0 .4} x^{0 .4}} < 0 \forall x \in (0, 1)$
Hence, f(x) is decreasing. Thus, $f (x)_{max} = f (0) = 1$

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