draw-the-graph-of-f-x-1-x-2-for-0-x-1-

Question Number 56179 by problem solverd last updated on 11/Mar/19

draw the graph of

f (x) = \sqrt{1 - x^{2}}

for 0 ⩽ x ⩽ 1

Commented by mr W last updated on 12/Mar/19

y = \sqrt{1 - x^{2}}

\Rightarrow x^{2} + y^{2} = 1^{2}

t h e g r a p h i s j u s t a p a r t o f a c i r c l e w i t h

r a d i u s 1 .

Answered by tanmay.chaudhury50@gmail.com last updated on 11/Mar/19

w i t h t h e u s e o f g r a p h i c a l a p p y o u c a n s e e t h e

g r a p h . .

t r y i n g t o s o l v e \dots

f (0) = 1

f (1) = 0

\frac{d f}{d x} = \frac{1}{2 \sqrt{1 - x^{2}}} \times - 2 x = \frac{- x}{\sqrt{1 - x^{2}}}

\frac{d^{2} f}{{d x}^{2}} = - 1 \times {\frac{\sqrt{1 - x^{2}} \times \frac{d x}{d x} - x \times \frac{d}{d x} (\sqrt{1 - x^{2}})}{1 - x^{2}}}

\frac{d^{2} f}{{d x}^{2}} = - 1 \times {\frac{\sqrt{1 - x^{2}} - x \times \frac{1}{2 \sqrt{1 - x^{2}}} \times - 2 x}{1 - x^{2}}}

= - 1 \times {\frac{\sqrt{1 - x^{2}} + \frac{x^{2}}{\sqrt{1 - x^{2}}}}{(1 - x^{2})}}

= \frac{- 1}{{(1 - x^{2})}^{\frac{3}{2}}}

\frac{d^{2} f}{{d x}^{2}} < 0 i n x [0, 1]

n o w i f \frac{d^{2} f}{{d x}^{2}} > 0 h o l d w a t e r t h a t m e a n s c o n c a v e u p

i f \frac{d^{2} f}{{d x}^{2}} < 0 s p i l l w a t e r t h a t m e a n s c o c a v e d o w n t h a t m e a n s C O N V E X

s o i n x [0, 1] s h a p e o f f (x) i s c o n v e x

. .

Commented by tanmay.chaudhury50@gmail.com last updated on 11/Mar/19

Commented by tanmay.chaudhury50@gmail.com last updated on 11/Mar/19