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Application of Derivatives

Find the intervals in which the following functions are strictly increasing or decreasing:
x²+ 2x – 5

Let $straight f left parenthesis straight x right parenthesis space equals space straight x squared plus 2 straight x minus 5 space space space space space space space space space space space therefore space space space space space straight f apostrophe left parenthesis straight x right parenthesis space equals space 2 straight x plus 2$
Now, $straight f apostrophe left parenthesis straight x right parenthesis greater than 0 space space when space 2 straight x plus 2 greater than 0 space space space straight i. straight e. space space space when space straight x greater than negative 1$
$therefore space space space space straight f left parenthesis straight x right parenthesis space is space increasing space in space left parenthesis negative 1 comma space infinity right parenthesis.$
Also, $straight f apostrophe left parenthesis straight x right parenthesis less than 0 space when space 2 straight x plus 2 less than 0 space space space space straight i. straight e. space when space straight x space less than space minus space 1$
$therefore space space space space straight f left parenthesis straight x right parenthesis space is space decreasing space in space left parenthesis negative infinity comma space minus 1 right parenthesis.$

127 Views

Find the intervals in which the following functions are strictly increasing or decreasing:
10 – 6x – 2x²

Let $straight f left parenthesis straight x right parenthesis space equals space 10 minus 6 straight x minus 2 straight x squared space$ $therefore space space space straight f apostrophe left parenthesis straight x right parenthesis space equals space minus 6 minus 4 straight x$
Now, $straight f apostrophe left parenthesis straight x right parenthesis greater than 0 space space space space space when space space space minus 6 minus 4 straight x space greater than 0 space space space space space straight i. straight e. space when space straight x space less than negative 3 over 2.$
$therefore space space space space space straight f left parenthesis straight x right parenthesis space is space increasing space in space open parentheses negative infinity comma space minus 3 over 2 close parentheses.$
$Also space straight f apostrophe left parenthesis straight x right parenthesis space less than space 0 space space space when space minus 6 minus 4 straight x space less than 0 space space space straight i. straight e. space when space straight x space greater than space minus 3 over 2. therefore space space space space straight f left parenthesis straight x right parenthesis space is space decreasing space in space open parentheses negative 3 over 2 comma space infinity close parentheses.$

105 Views

Which of the following function are strictly decreasing on

open parentheses 0 comma space straight pi over 2 close parentheses space ?

(a) cos x (b) cos 2x (c) cos 3x (d) tan x

(a) Let $straight f left parenthesis straight x right parenthesis space equals space cos space straight x comma space space space space space space space space therefore space space space straight f apostrophe left parenthesis straight x right parenthesis space equals space minus sin space straight x space$
In $open parentheses 0 comma space straight pi over 2 close parentheses comma space space straight f apostrophe left parenthesis straight x right parenthesis space equals space minus sin space straight x space less than space 0$
$therefore space space space space straight f left parenthesis straight x right parenthesis space equals space cosx space is space strictly space decreasing space on space open parentheses 0 comma space straight pi over 2 close parentheses$

(b) Let $straight f left parenthesis straight x right parenthesis space equals space cos space 2 straight x comma space space space space space space therefore space space space space space straight f apostrophe left parenthesis straight x right parenthesis space equals space minus 2 space sin space 2 straight x$
In $open parentheses 0 comma space straight pi over 2 close parentheses comma space space space straight f apostrophe left parenthesis straight x right parenthesis space equals space minus 2 space sin space 2 straight x space less than space 0$
$therefore space space space space space straight f left parenthesis straight x right parenthesis space equals space cos space 2 straight x$ is strictly decreasing on $open parentheses 0 comma space straight pi over 2 close parentheses.$
(c) Let f(x) = cos 3x, $therefore space space space straight f apostrophe left parenthesis straight x right parenthesis space equals space minus 3 space sin space 3 straight x$
$0 less than straight x less than straight pi over 2$
$rightwards double arrow space space space space space space 0 less than space 3 straight x space less than space fraction numerator 3 straight pi over denominator 2 end fraction$
Now $straight f apostrophe left parenthesis straight x right parenthesis space less than 0 space space space space when space 0 space less than space 3 straight x space less than space straight pi$
and $straight f apostrophe left parenthesis straight x right parenthesis space greater than 0 space space space when space straight pi space less than space 3 straight x space less than fraction numerator 3 straight pi over denominator 2 end fraction$
$therefore space space space space space straight f left parenthesis straight x right parenthesis space is space not space straight a space strictly space decreasing space function space on space open parentheses 0 comma space space straight pi over 2 close parentheses$

(d) Let $straight f left parenthesis straight x right parenthesis space equals tanx comma space space space space space space space space therefore space space space space space straight f apostrophe left parenthesis straight x right parenthesis space equals space sec squared straight x$
Now, $straight f apostrophe left parenthesis straight x right parenthesis greater than 0 space space for all space straight x space space space space element of space space space open parentheses 0 comma space space straight pi over 2 close parentheses$
$therefore space space space straight f left parenthesis straight x right parenthesis space is space not space straight a space strictly space decreasing space function space on space open parentheses 0 comma space space space straight pi over 2 close parentheses$

79 Views

Find the intervals in which the following functions are strictly increasing or decreasing:
6 – 9x – x ²

Let f(x) = 6 – 9x – x ^{2 $therefore space space space space straight f apostrophe left parenthesis straight x right parenthesis space equals space minus 9 minus 2 straight x$}Now, $straight f apostrophe left parenthesis straight x right parenthesis greater than 0 space space space when space minus 2 straight x minus 9 greater than 0 space space straight i. straight e. space space space when space straight x space less than negative 9 over 2$
$therefore space space space straight f left parenthesis straight x right parenthesis space is space increasing space for space straight x less than negative 9 over 2 therefore space space space space straight f left parenthesis straight x right parenthesis space is space increasing space in space open parentheses negative infinity comma space space fraction numerator negative 9 over denominator 2 end fraction close parentheses. Also comma space straight f apostrophe left parenthesis straight x right parenthesis space less than space 0 space space when space minus 2 straight x minus 9 less than 0 space space space straight i. straight e. space when space straight x greater than negative 9 over 2 therefore space space space space straight f left parenthesis straight x right parenthesis space is space increasing space for space straight x space greater than negative 9 over 2 therefore space space space space space space straight f left parenthesis straight x right parenthesis space is space decreasing space in space open parentheses negative 9 over 2 comma space space infinity close parentheses.$