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Application Of Derivatives

Question
CBSEENMA12035382
Wired Faculty App

Find the maximum or minimum values, if any, of the following functions without using the derivatives:
16x2 – 16x + 28

 

Solution

Let f(x) = 16x2 – 16x + 28 = 16(x2 - x) + 28
              equals 16 open parentheses straight x squared minus straight x plus 1 fourth close parentheses space plus space left parenthesis 28 minus 4 right parenthesis space equals space 16 open parentheses straight x minus 1 half close parentheses squared plus 24
Now,     open parentheses straight x minus 1 half close parentheses squared space greater or equal than space 0 space space space for all space space space straight x space space space element of space space straight R space space space space space space rightwards double arrow space space space space space space 16 open parentheses straight x minus 1 half close parentheses squared space greater or equal than space 0 space space space for all space space space straight x space space element of space space straight R
rightwards double arrow space 16 space open parentheses straight x minus 1 half close parentheses squared plus 24 space greater or equal than space 24 space space for all space space straight x space space space element of space space straight R space space space rightwards double arrow space space space space straight f left parenthesis straight x right parenthesis space greater or equal than space 24 space for all space space straight x space space space element of space space straight R
therefore minimum value of f(x) is 24. It has no maximum value.

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