First slide

Rate Measure

Question

The minimum rate of change of the function $f (x) = 3 x^{5} - 5 x^{3} + 5 x - 7$ is

Moderate

A

$\frac{3}{4}$
B

$\frac{5}{4}$
C

$\frac{2}{3}$
D

$\frac{3}{2}$

Solution

Rate of change of $f (x) is f^{'} (x)$
$= 15 x^{4} - 15 x^{2} + 5 = 15 (x^{4} - x^{2} + \frac{1}{3})$
$= 15 ({(x^{2} - \frac{1}{2})}^{2} + \frac{1}{12}) \geq \frac{15}{12} = \frac{5}{4}$
The minimum vale of $f$ is attained at $x = + \frac{1}{\sqrt{2}}$ and equals $\frac{5}{4}$ .

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