1. Find the maximum and minimum values, if any, of the following functions
given by 

(ii) f (x) = 9x^ 2 + 12x + 2

Answers (1)

Given function is,
f (x) = 9x^ 2 + 12x + 2
add and subtract 2 in given equation
f (x) = 9x^ 2 + 12x + 2 + 2- 2\\ f(x)= 9x^2 +12x+4-2\\ f(x)= (3x+2)^2 - 2
Now,
(3x+2)^2 \geq 0\\ (3x+2)^2-2\geq -2    for every x \ \epsilon \ R
Hence, minimum value occurs when 
(3x+2)=0\\ x = \frac{-2}{3}
Hence, the minimum value of function f (x) = 9x^2+12x+2 occurs at  x = \frac{-2}{3}
and the minimum value is 
f(\frac{-2}{3}) = 9(\frac{-2}{3})^2+12(\frac{-2}{3})+2=4-8+2 =-2 \\
            
and it is clear that there is no maximum value of  f (x) = 9x^2+12x+2

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