Application of Derivatives

Find intervals in which the function given by is (a) strictly increasing (b) strictly decreasing. - Wired Faculty

Question

Find intervals in which the function given by
$straight f left parenthesis straight x right parenthesis space equals space 3 over 10 straight x to the power of 4 minus 4 over 5 straight x cubed minus 3 straight x squared plus 36 over 5 straight x plus 11$
is (a) strictly increasing (b) strictly decreasing.

Solution

straight f left parenthesis straight x right parenthesis space equals space 3 over 10 straight x to the power of 4 minus 4 over 5 straight x cubed minus 3 straight x squared plus 36 over 5 straight x plus 11

therefore space space space space space straight f apostrophe left parenthesis straight x right parenthesis space equals space 3 over 10 cross times space 4 straight x cubed space minus 4 over 5 cross times space 3 straight x squared minus 3 space cross times space 2 straight x space plus space 36 over 5 space space space space space space space space space space space space space space space space space equals 6 over 5 straight x cubed minus 12 over 5 straight x squared minus 6 straight x plus 36 over 5 space equals space 6 over 5 left parenthesis straight x cubed minus 2 straight x squared minus 5 straight x plus 6 right parenthesis therefore space space space space straight f apostrophe left parenthesis straight x right parenthesis space equals space 6 over 5 left parenthesis straight x minus 1 right parenthesis thin space left parenthesis straight x plus 2 right parenthesis thin space left parenthesis straight x minus 3 right parenthesis

Now,

straight f apostrophe left parenthesis straight x right parenthesis space equals space 0 space space space space rightwards double arrow space space space space space space 6 over 5 left parenthesis straight x minus 1 right parenthesis thin space left parenthesis straight x plus 2 right parenthesis thin space left parenthesis straight x minus 3 right parenthesis space equals space 0

therefore space space space space space space left parenthesis straight x minus 1 right parenthesis thin space left parenthesis straight x plus 2 right parenthesis thin space left parenthesis straight x minus 3 right parenthesis space equals space 0 space space space space space space space rightwards double arrow space space space straight x space equals space 1 comma space space minus 2 comma space space 3

The points x = 1, -2, 3 divide the real line into four disjoint intervals

left parenthesis negative infinity comma space space minus 2 right parenthesis comma space space left parenthesis negative 2 comma space space 1 right parenthesis comma space space space left parenthesis 1 comma space space 3 right parenthesis comma space space left parenthesis 3 comma space infinity right parenthesis.

In space left parenthesis negative infinity comma space space minus 2 right parenthesis comma space space space straight f apostrophe left parenthesis straight x right parenthesis space equals space 6 over 5 left parenthesis negative right parenthesis thin space left parenthesis negative right parenthesis thin space left parenthesis negative right parenthesis space equals space minus ve space less than space 0

therefore space space space space space straight f space is space strictly space decreasing space in space left parenthesis negative infinity comma space minus 2 right parenthesis.

left parenthesis negative 2 comma space 1 right parenthesis comma space space straight f apostrophe left parenthesis straight x right parenthesis space equals space 6 over 5 left parenthesis negative right parenthesis thin space left parenthesis plus right parenthesis thin space left parenthesis negative right parenthesis space equals space plus ve space greater than space 0

therefore

f is strictly decreasing in (1, 3)
In

left parenthesis 1 comma space 3 right parenthesis comma space space space straight f apostrophe left parenthesis straight x right parenthesis space equals space 6 over 5 left parenthesis plus right parenthesis thin space left parenthesis plus right parenthesis thin space left parenthesis negative right parenthesis space equals space minus ve less than space 0

therefore space space space straight f space is space strictly space decreasing space in space left parenthesis 1 comma space 3 right parenthesis.

left parenthesis 3 comma space infinity right parenthesis comma space space straight f apostrophe left parenthesis straight x right parenthesis space equals space 6 over 5 left parenthesis plus right parenthesis thin space left parenthesis plus right parenthesis space left parenthesis plus right parenthesis space equals space plus ve space greater than space 0

therefore space space space space space straight f space is space strictly space increasing space in space left parenthesis 3 comma space infinity right parenthesis.