The minimum rate of change of the function f(x)=3×5-5×3+5x-7 is

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

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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