Is #f(x)=(-2x^3+x^2-2x-4)/(4x-2)# increasing or decreasing at #x=0#?

Answer 1

#"increasing at x = 0"#

#"to determine if f(x) is increasing/decreasing at x = a"# #"differentiate and evaluate at x = a"#
#• " if "f'(x)>0" then f(x) is increasing at x = a"#
#• " if "f'(x)<0" then f(x) is decreasing at x = a"#
#"differentiate using the "color(blue)"quotient rule"#
#"given "f(x)=(g(x))/(h(x))" then"#
#f'(x)=(h(x)g'(x)-g(x)h'(x))/(h(x))^2larrcolor(blue)"quotient rule"#
#g(x)=-2x^3+x^2-2x-4rArrg'(x)=-6x^2+2x-2#
#h(x)=4x-2rArrh'(x)=4#
#f'(x)=((4x-2)(-6x^2+2x-2)-4(-2x^3+x^2-2x-4))/(4x-2)^2#
#rArrf'(0)=((-2)(-2)-4(-4))/4=5>0#
#"since " f'(x)>0" then f(x) is increasing at x = 0"# graph{(-2x^3+x^2-2x-4)/(4x-2) [-10, 10, -5, 5]}
Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 2

To determine whether ( f(x) = \frac{-2x^3 + x^2 - 2x - 4}{4x - 2} ) is increasing or decreasing at ( x = 0 ), you can analyze the sign of the derivative of ( f(x) ) at that point.

To find the derivative, apply the quotient rule:

[ f'(x) = \frac{(4x - 2)(-6x^2 + 2x - 2) - (-2x^3 + x^2 - 2x - 4)(4)}{(4x - 2)^2} ]

Now, substitute ( x = 0 ) into ( f'(x) ) to determine whether the derivative is positive (indicating increasing), negative (indicating decreasing), or zero (indicating a possible local extremum) at ( x = 0 ).

[ f'(0) = \frac{(4 \times 0 - 2)(-6 \times 0^2 + 2 \times 0 - 2) - (-2 \times 0^3 + 0^2 - 2 \times 0 - 4)(4)}{(4 \times 0 - 2)^2} ]

[ f'(0) = \frac{(-2)(-2) - (-4)(4)}{(-2)^2} = \frac{4 - (-16)}{4} = \frac{20}{4} = 5 ]

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

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
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.

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.

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.

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.

Not the question you need?

Drag image here or click to upload

Or press Ctrl + V to paste
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7