Is #f(x)=(x-3)^3+3x^2-2x # increasing or decreasing at #x=0 #?

Answer 1

Evelyn Agard

Hence, #f# is increasing at #x=0.#

We use the criteria #f'(a)>0# #rArr# #f# is #uarr# at #a.# #f'(a)<0# #rArr# #f# is #darr# at #a#.

Now, #f(x)=(x-3)^3+3x^2-2x. rArr f'(x)=3(x-3)^2+6x-2 rArr f'(0)=3(-3)^2-2=25>0#

Hence, #f# is increasing at #x=0.#

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Answer 2

Christopher Allers

To determine whether ( f(x) = (x - 3)^3 + 3x^2 - 2x ) is increasing or decreasing at ( x = 0 ), we need to analyze the sign of the first derivative ( f'(x) ) at that point.

( f'(x) ) is the derivative of ( f(x) ) with respect to ( x ).

( f'(x) = 3(x - 3)^2 + 6x - 2 ).

Evaluate ( f'(0) ).

( f'(0) = 3(0 - 3)^2 + 6(0) - 2 ).

( f'(0) = 3(9) - 2 ).

( f'(0) = 27 - 2 = 25 ).

Since ( f'(0) = 25 > 0 ), the function is increasing at ( 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.