Find the absolute maximum and absolute minimum of f on the interval (-4, -1]:
f(x)=(x^3+8x^2+19x+12)/(x+4)
A. Maximum: None; Minimum: (-2, -1)
B. Maximum: (-4, 3); Minimum (-1, 0)
C. Maximum: (-4, 3); Minimum: (-2, -1)
D. Maximum: None; Minimum: (-1, 0)
E. none of these
the top factors, so
f(x) = (x+1)(x+3)(x+4)/(x+4)
= x^2 + 4x + 3 , x ≠ -4
f ' (x) = 2x + 4
= 0 for a max/min
2x + 4 = 0
x = -2
Since the function is basically the parabola
y = x^2 + 4x + 3 , with a hole at (-4,3)
which opens up,
x = -2 will produce a minimum
f(-2) = 4 - 8 + 3 = -1
so (-2,-1) is a minimum
choice A looks good
To find the absolute maximum and minimum of the function f(x) on the interval (-4, -1], we can follow these steps:
Step 1: Find the critical points of f(x) by setting the derivative equal to zero and solving for x.
The derivative of f(x) can be found using the quotient rule:
f'(x) = [(x+4)(3x^2 + 16x + 19) - (x^3 + 8x^2 + 19x + 12)(1)] / (x+4)^2
= (3x^3 + 13x^2 + 19x + 76) / (x+4)^2
Setting f'(x) equal to zero:
(3x^3 + 13x^2 + 19x + 76) = 0
Using synthetic division or other methods, we find that x = -2 is a critical point.
Step 2: Check the function at the critical point and the endpoints of the interval.
Evaluate f(x) at the critical point x = -2:
f(-2) = ((-2)^3 + 8(-2)^2 + 19(-2) + 12) / (-2 + 4)
= (-8 + 32 - 38 + 12) / 2
= -2
Evaluate f(x) at the endpoints x = -4 and x = -1:
f(-4) = ((-4)^3 + 8(-4)^2 + 19(-4) + 12) / (-4 + 4)
= (-64 + 128 - 76 + 12) / 0
= undefined
f(-1) = ((-1)^3 + 8(-1)^2 + 19(-1) + 12) / (-1 + 4)
= (-1 + 8 - 19 + 12) / 3
= 0
Step 3: Determine the absolute maximum and minimum.
The function f(x) is undefined at x = -4, so it does not have an absolute maximum or minimum at x = -4.
The function f(x) has a minimum at x = -1 with a value of 0.
Therefore, the correct answer is:
D. Maximum: None; Minimum: (-1, 0)
To find the absolute maximum and absolute minimum of the function f(x) on the interval (-4, -1], you can follow these steps:
1. Determine the critical points of the function within the given interval by finding the values of x for which f'(x) = 0 or f'(x) is undefined.
To do this, first calculate the derivative of f(x) using the quotient rule:
f'(x) = [(x+4)(3x^2 + 16x + 19) - (x^3+8x^2+19x+12)(1)] / (x+4)^2
Simplifying the above expression, we get:
f'(x) = (2x^2 + 11x + 16)/(x+4)^2
To find the critical points, set f'(x) = 0 and solve for x:
2x^2 + 11x + 16 = 0
Solving the quadratic equation, we get:
x = (-11 ± √(11^2 - 4(2)(16))) / (2(2))
x = (-11 ± √(121 - 128)) / 4
x = (-11 ± √(-7)) / 4
Since the discriminant is negative, the quadratic equation does not have real solutions. Therefore, there are no critical points within the interval (-4, -1].
2. Next, check the values of the function at the endpoints of the interval (-4, -1].
For x = -4:
f(-4) = (-4^3 + 8(-4)^2 + 19(-4) + 12) / (-4 + 4)
= (-64 + 128 - 76 + 12) / 0 (undefined)
For x = -1:
f(-1) = (-1^3 + 8(-1)^2 + 19(-1) + 12) / (-1 + 4)
= (-1 + 8 - 19 + 12) / 3
= 0 / 3
= 0
3. Finally, compare the values obtained above to determine the absolute maximum and absolute minimum.
Since the function is undefined at x = -4, it cannot have an absolute maximum or minimum at that point.
The function value at x = -1 is 0, which is the absolute minimum.
Therefore, the answer is D. Maximum: None; Minimum: (-1, 0).