How to check the sign of the derative of a function in its given interval?t

Asked by Kishujaiswal353 | 21st Sep, 2014, 10:55: AM

Expert Answer:

1. space F i n d space t h e space d e r i v a t i v e space o f space t h e space g i v e n space f u n c t i o n. 2. space I f space t h e space i n t e r v a l space i s space g i v e n comma space t h e n space t a k e space a n y space r e a l space n o. space i n space b e t w e e n space t h e space i n t e r v a l space a n d space s u b s t i t u t e space t h e space v a l u e space i n space fraction numerator d y over denominator d x end fraction. 3. space C h e c k space w h e t h e r space t h e space v a l u e space c o m e s space o u t space t o space b e space p o s i t i v e space o r space n e g a t i v e. 4. space I f space t h e space v a l u e space o f space fraction numerator d y over denominator d x end fraction space i s space p o s i t i v e comma space t h e n space t h e space f u n c t i o n space i s space a n space i n c r e a sin g space f u n c t i o n space a n d space i f space t h e space v a l u e space o f space fraction numerator d y over denominator d x end fraction space i s space n e g a t i v e comma t h e n space t h e space f u n c t i o n space i s space a space d e c r e a sin g space f u n c t i o n. F o r space e x a m p l e comma space L e t space f left parenthesis x right parenthesis space equals space x cubed minus 2 x squared plus 7. space T h e space f u n c t i o n space n e e d s space t o space b e space c h e c k e d space b e t w e e n space t h e space i n t e r v a l space left square bracket 1 comma 3 right square bracket f apostrophe left parenthesis x right parenthesis space equals space 3 x squared minus 4 x T a k i n g space a space v a l u e space i n space b e t w e e n space 1 space & space 3 comma space w h i c h space c a n space b e space 2. f apostrophe left parenthesis 2 right parenthesis equals 3 left parenthesis 2 right parenthesis squared minus 4 left parenthesis 2 right parenthesis equals 12 minus 8 equals 4 greater than 0 T h e r e f o r e space t h e space f u n c t i o n space i s space i n c r e a sin g space i n space t h e space i n t e r v a l space left square bracket 1 comma 3 right square bracket   space

Answered by Prasenjit Paul | 21st Sep, 2014, 10:03: PM