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

How do you find the local extremas for #f(x)=x^(1/3)(x+8)#?

Answer 1

Has a local minimum at #x = -2#

The critical points are obtained by solving #d/(dx)f(x)=(4 (2 + x))/(3 x^(2/3))=0# Solving for #x# we get #x = -2# The critical point qualification is done by calculating #d^2/(dx)^2 f(x) = (4 (-4 + x))/(9 x^(5/3))# So #d^2/(dx)^2 f(-2) = 0.839947# Then the critical point is a minimum. If you where using a symbolic processor be aware of #x^{1/3} equiv x/(abs x)abs x^{1/3}# and #x^{5/3} equiv x/(abs(x))abs[x]^(5/3)#

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Answer 2

Paisley Johnson

To find the local extrema for ( f(x) = x^{1/3}(x+8) ), follow these steps:

Find the first derivative of the function ( f(x) ) using the product rule.
Set the first derivative equal to zero and solve for ( x ).
Evaluate the second derivative of ( f(x) ) at each critical point to determine the concavity.
Determine whether each critical point corresponds to a local minimum, local maximum, or neither, based on the first and second derivative tests.

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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.