Home > Calculus > Classifying Critical Points and Extreme Values for a Function

How do you determine where the function is increasing or decreasing, and determine where relative maxima and minima occur for # f(x) = x² + 2x - 3#?

Answer 1

Use the derivative. The function is (strictly) increasing over intervals where #f'(x) > 0# and (strictly) decreasing over intervals where #f'(x) < 0#. Local extreme points occur at critical points where #f'(x) = 0# or where #f'(x)# is undefined.

If #f(x)=x^2+2x-3#, then #f'(x)=2x+2# so that #f'(x) < 0# when #x < -1# and #f'(x) > 0# when #x > -1#. This means #f# is decreasing over the interval #x <= -1# and #f# is increasing over the interval #x >= -1#. #f# has a local minimum value at the critical point #x=-1# equal to #f(-1)=1-2-3=-4#.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.