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 $f\left(x\right)=3{x}^{5}-5{x}^{3}+5x-7$ is

1. A

$\frac{3}{4}$

2. B

$\frac{5}{4}$

3. C

$\frac{2}{3}$

4. D

$\frac{3}{2}$

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### Solution:

Rate of change of

$=15{x}^{4}-15{x}^{2}+5=15\left({x}^{4}-{x}^{2}+\frac{1}{3}\right)$

$=15\left({\left({x}^{2}-\frac{1}{2}\right)}^{2}+\frac{1}{12}\right)\ge \frac{15}{12}=\frac{5}{4}$

The minimum vale of $f$ is attained at $x=+\frac{1}{\sqrt{2}}$ and equals $\frac{5}{4}$.

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