A Note As f is twice differentiable at c , we mean second order derivative of f exists at c . Example Find local minimum value of the function f given by f ( x ) = + | x |, x ∈ R . Solution Note that the given function is not differentiable at x = . So, second derivative test fails.
Let us try first derivative test. Note that is a critical point of f . Now to the left of , f ( x ) = – x and so f ′ ( x ) = – < . Also to the right of , f ( x ) = + x and so f ′ ( x ) = > .
Therefore, by first derivative test, x = is a point of local minima of f and local minimum value of f is f ( ) = . Example Find local maximum and local minimum values of the function f given by f ( x ) = x + x – x + Solution We have f ( x ) = x + x – x + f ′ ( x ) = x + x – x = x ( x – ) ( x + ) f ′ ( x ) = at x = , x = and x = – . Now f ″ ( x ) = x + x – = ( x + x – ) ′′ = − ′′ > ′′ − > Therefore, by second derivative test, x = is a point of local maxima and local maximum value of f at x = is f ( ) = while x = and x = – are the points of local minima and local minimum values of