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Is #f(x)=(x-2)(x+5)(x-1)# increasing or decreasing at #x=-1#?

Answer 1

#f(x)# is decreasing when #x=-1#

To determine whether #f(x)# is increasing or decreasing at #x=-1#, we have to check whether #f'(x)# is positive or negative at #x=-1#.

First, find #f'(x)# using product rule: #f(x) = (x-2)(x+5)(x-1)# #f'(x) = cancel((1))(x+5)(x-1) + (x-2)cancel((1))(x-1) + (x-2)(x+5)cancel((1))# #f'(x) = 3x^2+4x-13#

Next, find #f'(-1)#: #f'(-1)=-14#

Since #f'(-1)# is negative, that means that #f(x)# is decreasing when #x=-1#.

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Answer 2

Charles Anctil

To determine whether ( f(x) = (x - 2)(x + 5)(x - 1) ) is increasing or decreasing at ( x = -1 ), we can use the first derivative test.

Find the derivative of ( f(x) ): [ f'(x) = (x + 5)(x - 1) + (x - 2)(x - 1) + (x - 2)(x + 5) ] [ f'(x) = (x^2 + 4x - 5) + (x^2 - 3x + 2) + (x^2 + 3x - 10) ] [ f'(x) = 3x^2 + 4x - 3x^2 - 3x + 2x^2 - 7 ] [ f'(x) = 2x^2 + x - 7 ]
Evaluate ( f'(-1) ): [ f'(-1) = 2(-1)^2 + (-1) - 7 = 2 - 1 - 7 = -6 ]

Since ( f'(-1) < 0 ), the function ( f(x) ) is decreasing at ( x = -1 ).

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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.