3. Find the local maxima and local minima, if any, of the following functions. Find
also the local maximum and the local minimum values, as the case may be:

(vii) g (x) = \frac{1}{x^2 + 2}

Answers (1)

Gien function is 
g (x) = \frac{1}{x^2 + 2}\\ g^{'}(x) = \frac{-2x}{(x^2+2)^2}\\ g^{'}(x) = 0\\ \frac{-2x}{(x^2+2)^2} = 0\\ x = 0
Hence., x = 0 is only critical point 
Now, we use the second derivative test
g^{''}(x) = -\frac{-2(x^2+2)^2-(-2x){2(x^2+2)(2x)}}{((x^2+2)^2)^2} \\ g^{''}(0) = \frac{-2\times4}{(2)^4} = \frac{-8}{16} = -\frac{1}{2}< 0
Hence, 0 is the point of local maxima and the maximum value is 
g (0) = \frac{1}{0^2 + 2} = \frac{1}{2}

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