Is #f(x)=(x2)(x+5)(x1)# increasing or decreasing at #x=1#?
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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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When evaluating a onesided 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.
When evaluating a onesided 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.
When evaluating a onesided 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.
When evaluating a onesided 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.
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