If X^7-X^3=1542, then what is the value of x?
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Answer:
Given : x^7 -x^3 = 1542x
7
−x
3
=1542
Solution : f(x) = x^7 -x^3x
7
−x
3
then
f'(x) =
\begin{gathered}7x^6-3x^2\\\\x^2(7x^4-3)\end{gathered}
7x
6
−3x
2
x
2
(7x
4
−3)
This shows that f has an inflection point at 0
then
it is increasing over (-\infty, - \sqrt[4]{\frac{3}{7}})(−∞,−
4
7
3
)
decreasing over [-\sqrt[4]{\frac{3}{7}},\sqrt[4]{\frac{3}{7}}][−
4
7
3
,
4
7
3
]
increasing over [\sqrt[4]{\frac{3}{7}}, \infty)[
4
7
3
,∞)
Thus ,
The local maximum is
f(-\sqrt[4]{\frac{3}{7}})= \frac{4}{7}(\sqrt[4]{\frac{3}{7}})^3 < 1f(−
4
7
3
)=
7
4
(
4
7
3
)
3
<1
i.e
it has no solution
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