Home > Calculus > Interpreting the Sign of the First Derivative (Increasing and Decreasing Functions)

Is #f(x)=-x^3+3x^2-x+2# increasing or decreasing at #x=-1#?

Answer 1

Scarlett Allison

decreasing at x = - 1

Take into consideration whether f(x) is increasing or decreasing at x = a.

• At x = a, f(x) is increasing if f'(a) > 0.

• At x = a, f(x) is decreasing if f'(a) < 0.

differentiate using the #color(blue)"power rule"#

#rArrf'(x)=-3x^2+6x-1#

#rArrf'(-1)=-3(-1)^2+6(-1)-1=-10#

f(x) is decreasing at x = -1 because f'(-1) < 0. The graph is {-x^3+3x^2+6x-1 [-50.76, 50.76, -25.4, 25.36]}.

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Answer 2

Paisley Johnson

To determine if ( f(x) = -x^3 + 3x^2 - x + 2 ) is increasing or decreasing at ( x = -1 ), we need to evaluate the sign of the derivative of ( f(x) ) at ( x = -1 ).

First, find the derivative of ( f(x) ):

( f'(x) = -3x^2 + 6x - 1 )

Now, evaluate ( f'(-1) ):

( f'(-1) = -3(-1)^2 + 6(-1) - 1 )

( f'(-1) = -3 + (-6) - 1 )

( f'(-1) = -3 - 6 - 1 )

( f'(-1) = -10 )

Since ( f'(-1) = -10 ) is negative, 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.