Application of Derivatives

Question

Find the maximum and minimum values, if any, of the following functions without using the derivatives:
4x² – 4x + 11

Solution

$Let space space space straight f left parenthesis straight x right parenthesis space equals space 4 straight x squared minus 4 straight x plus 11 space space equals space 4 left parenthesis straight x squared minus straight x right parenthesis space plus 11 space equals space 4 space open parentheses straight x squared minus straight x plus 1 fourth close parentheses plus left parenthesis 11 minus 1 right parenthesis space space space space space space space space space equals 4 space open parentheses straight x minus 1 half close parentheses squared plus 10$
Now, $open parentheses straight x minus 1 half close parentheses squared space greater or equal than space space space 0 space space space space space for all space space straight x space space space element of space space space straight R$

therefore space space space 4 open parentheses straight x minus 1 half close parentheses squared space greater or equal than space space 0 space space space space space for all space straight x space space space element of space space straight R space space space space space space space space space space space space space space space rightwards double arrow space space space space space space 4 space open parentheses straight x minus 1 half close parentheses squared plus 10 space greater or equal than space 10 space space space space for all space space straight x space element of space space straight R rightwards double arrow space space space space straight f left parenthesis straight x right parenthesis space greater or equal than space 10 space space space space space for all space space straight x space space space element of space space space straight R therefore space space space minimum space value space of space straight f left parenthesis straight x right parenthesis space is space 10. space It space has space no space maximum space value.

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