# Is #f(x)=(3x^3-2x^2-2x+5)/(x+2)# increasing or decreasing at #x=3#?

and

increasing

given

proceed by splitting up

to acquire

To obtain the first derivative, find

evaluate

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

To determine whether ( f(x) = \frac{{3x^3 - 2x^2 - 2x + 5}}{{x + 2}} ) is increasing or decreasing at ( x = 3 ), we can use the first derivative test. First, find the derivative of ( f(x) ) with respect to ( x ) using the quotient rule. Then, evaluate the derivative at ( x = 3 ). If the derivative is positive, the function is increasing at that point. If it's negative, the function is decreasing.

( f'(x) = \frac{{(9x^2 - 4x - 2)(x + 2) - (3x^3 - 2x^2 - 2x + 5)}}{{(x + 2)^2}} )

After simplifying, ( f'(x) = \frac{{3x^3 - 2x^2 - 2x + 5 - 9x^2 - 18x - 4x - 8}}{{(x + 2)^2}} )

( f'(x) = \frac{{3x^3 - 11x^2 - 24x - 3}}{{(x + 2)^2}} )

Evaluate ( f'(3) ):

( f'(3) = \frac{{3(3)^3 - 11(3)^2 - 24(3) - 3}}{{(3 + 2)^2}} )

( f'(3) = \frac{{81 - 99 - 72 - 3}}{{25}} )

( f'(3) = \frac{{-93}}{{25}} )

Since ( f'(3) ) is negative, the function ( f(x) ) is decreasing at ( x = 3 ).

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

- How do you find the critical points of the function #f(x) = x / (x^2 + 4)#?
- What are the values and types of the critical points, if any, of #f(x, y) = x^3+y^3-3*x*y-7#?
- What are the global and local extrema of #f(x)=x^3-x^2-x# ?
- Using mean value theorem show that: #x< sin^-1x#, for #x>0#?
- Is #f(x)= cos(x+(5pi)/4) # increasing or decreasing at #x=-pi/4 #?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7