Application of Derivatives

Question

Without using derivatives, find the maximum or minimum value of the function 9x²+ 12x + 2.

Solution

$Let space straight f left parenthesis straight x right parenthesis space equals space 9 straight x squared plus 12 straight x plus 2 space equals space left parenthesis 9 straight x squared plus 12 straight x right parenthesis space plus 2$
$equals space 9 open parentheses straight x squared plus 4 over 3 straight x close parentheses space plus 2 space equals space 9 space open parentheses straight x squared plus 4 over 3 straight x plus 4 over 9 close parentheses plus left parenthesis 2 minus 4 right parenthesis space equals space 9 open parentheses straight x plus 2 over 3 close parentheses squared minus 2$

Now,

open parentheses straight x plus 2 over 3 close parentheses squared space greater or equal than space space 0 space space space for all space straight x space space element of space space straight R space space space space space rightwards double arrow space space space space 9 open parentheses straight x plus 2 over 3 close parentheses squared space greater or equal than space 0 space for all space straight x space space element of space space straight R

therefore space space space space space 9 open parentheses straight x plus 2 over 3 close parentheses squared minus 2 space greater or equal than space minus 2 space for all space space straight x space space element of space space straight R therefore space space space space space straight f left parenthesis straight x right parenthesis space greater or equal than negative 2 space space space space for all space space straight x space space element of space space straight R therefore space space minimum space value space of space straight f left parenthesis straight x right parenthesis space is space minus 2 space and space is space obtained space at space straight x space equals space minus 2 over 3. There space is space no space maximum space value space of space the space function. space

For Daily Free Study Material Join wiredfaculty