Question

Define 'a' if f(x) = x^10 - x^3 + 1 is monotonically increasing on [a, infinity] ?

Answer 1

$f(x) = x^{10}-x^3+1$
is monotonically increasing . it means
f'(x) ≥ 0 in interval [a, ∞ )
so,
$f'(x) =10x^{10-1}-3x^{3-1}+0\\\\=10x^9-3x^2\\\\=x^2(10x^7-3)$

now,
$f'(x)=x^2(10x^7-3) \geqslant 0 \\ \\ x \geqslant \sqrt[7]{ \frac{3}{10} } \: \: \: \: \: or \: \: \: \: \: \: x \leqslant 0$
hence,
$a=(\frac{3}{10})^{\frac{1}{7}}$