What is the nearest point in f(x) to (5,2) where f(x)=−0.5x^2 +3


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Question Number 192132 by Red1ight last updated on 09/May/23

$What is the nearest point in f (x) to (5, 2)$ $where f (x) = - 0.5 x^{2} + 3$
	Answered by mehdee42 last updated on 09/May/23

	$a c c o r d i n g t o t h e d i a g r a m :$ $l e t : H (h, f (h))$ $f^{'} (x) = - x \Rightarrow m_{l_{1}} = - h & m_{l_{2}} = \frac{2 - f (h)}{5 - h}$ $m_{l_{1}} \times m_{l_{2}} = - 1 \Rightarrow \frac{2 - 0.5 h^{2} - 3}{5 - h} \times (- h) = - 1 \Rightarrow h = \sqrt[3]{10}$ $\Rightarrow H (\sqrt[3]{10}, 3 - 0.5 \sqrt[3]{100}) ✓$
	Commented by mehdee42 last updated on 09/May/23

	Commented by Red1ight last updated on 09/May/23

	$why m_{l_{1}} = - h ?$
	Commented by mehdee42 last updated on 10/May/23

	$t h e l i n e l_{1} i s t a n g e n t t o t h e c o r v e a t p o i n t H .$ $s o i t s s l o p e i s e q u a l t o t h e v a l u e o f t h e$ $d e r i v a t i v e o f t h e f u n c t i o n p e r l e n g t h o f$ $t h e t a n g e n t p o i n t . {t h a t}^{'} s m e a n m_{l_{1}} = f^{'} (h) = - h$ $o k ?$
	Commented by Red1ight last updated on 09/May/23

	$Thanks sir$
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