Home > Calculus > Identifying Turning Points (Local Extrema) for a Function

How do you find the local extremas for #g(x) = - |x+6|#?

Answer 1

Differentiability is a stronger condition than continuity

# { ("Differentiability", =>, "Continuity"), ("Continuity", NOT =>, "Differentiability" ) :} #

You cannot use calculus as although the function is continuous everywhere, it is not differentiable everywhere, and specifically it is not differentiable at the extrema that we seek (which happens to be a maximum)

But we can plot the function #y=-|x+6| #

graph{-|x+6| [-10, 10, -5, 5]}

And observer that there is a maximum at #(-6,0)#

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Answer 2

Samantha Adkison

To find the local extrema of \(g(x) = -|x+6|\), first find the critical points by setting the derivative of \(g(x)\) equal to zero. The critical points occur where the absolute value function changes sign, which is at \(x = -6\). Since \(g(x)\) is piecewise, you need to check the sign of the derivative on both sides of \(x = -6\) to determine if it's a maximum, minimum, or neither. The derivative of \(g(x)\) is \(g'(x) = -\text{sign}(x+6)\), which is \(0\) for \(x=-6\), and \(g'(x) < 0\) for \(x < -6\), and \(g'(x) > 0\) for \(x > -6\). Therefore, \(x = -6\) is a local maximum for \(g(x) = -|x+6|\).

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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.