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How do you determine the concavity for #f(x)=x^4# ?

Answer 1

#f(x)=x^4# is concave upward on #(-infty,+infty)#.

By taking the derivatives, #f'(x)=4x^3# #f''(x)=12x^2 geq 0#, which means that #f'# is increasing on #(-infty,+infty)#, which means that #f# is concave upward on #(-infty,+infty)#.

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Answer 2

Christopher Allers

To determine the concavity of ( f(x) = x^4 ), you need to find the second derivative and then evaluate it. The concavity can be determined by analyzing the sign of the second derivative.

First, find the first derivative of ( f(x) ): [ f'(x) = 4x^3 ]

Then, find the second derivative: [ f''(x) = 12x^2 ]

Now, evaluate the second derivative at critical points (where ( f''(x) = 0 )) and determine the intervals where it is positive or negative. Since ( f''(x) = 12x^2 ), it is positive for ( x > 0 ) and negative for ( x < 0 ).

Therefore, the concavity of ( f(x) = x^4 ) is:

Concave up for ( x > 0 )
Concave down for ( x < 0 )

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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.