Can-we-exactly-find-r-max-a-r-2-2rt-a-t-2-ar-t-2-t-is-parameter-

Question Number 212500 by ajfour last updated on 15/Oct/24

C a n w e e x a c t l y f i n d r_{m a x} (a) .

r^{2} + \frac{2 r t}{a} (t + 2 \sqrt{a r}) = t^{2} \forall t i s p a r a m e t e r

Answered by mr W last updated on 15/Oct/24

(\frac{2 r}{a} - 1) t^{2} + 4 r \sqrt{\frac{r}{a}} t + r^{2} = 0

Δ = \frac{4 r^{3}}{a} - r^{2} (\frac{2 r}{a} - 1) ⩾ 0

(\frac{2 r}{a} + 1) r^{2} ⩾ 0

\frac{2 r}{a} + 1 ⩾ 0

t h i s i s a l w a y s t r u e, s i n c e a r ⩾ 0 .

i f a > 0 : r ⩾ 0

i f a < 0 : r ⩽ 0

Commented by ajfour last updated on 16/Oct/24

T r u e! i t h i n k r_{m a x} = a / 2

Commented by mr W last updated on 16/Oct/24

Commented by mr W last updated on 16/Oct/24

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