Find the Absolute Maximum and Absolute Minimum of f on (0,3].
f(x)=(x^3-4x^2+7x)/x
Multiple choice question
I know the minimum is (2,3) but the maximum is either nothing or (0,7) but I can't tell which one
on the given domain,
f(x) = x^2-4x+7
minimum is f(2)=3
limit as x->0 of f(x) = 7
f(3) = 4
So,
abs min = -9
abs max = 7
not strictly true, since f(0) is undefined, and there is no maximum number less than 7.
See the graph at
so, I guess the real(!) answer is indeed that there is no absolute max value.
Well, you're in a bit of a pickle! Trying to find the maximum and minimum can be quite precarious, especially when dealing with fractions and undefined points.
Let's break it down, shall we? We know that f(x) = (x^3 - 4x^2 + 7x) / x.
First, let's examine the situation for x = 0. Uh-oh! We run into a bit of trouble here because x cannot be zero since it's in the interval (0,3]. So we need to exclude zero from our analysis.
Next, let's consider the points near zero. As x approaches zero from the right side, the function becomes larger and larger, approaching positive infinity. However, it doesn't actually have an absolute maximum because f(x) just keeps growing! So, (0,7) isn't a valid maximum point since it's not actually attainable in this case.
Therefore, we can conclude that the function f(x) does not have an absolute maximum on the interval (0,3].
As for the absolute minimum, you're correct! At x = 2, f(x) reaches its minimum value of 3. So, the absolute minimum occurs at (2,3).
Keep in mind that mathematical analysis can sometimes be a juggling act, but I'm here to bring some laughter and clarity to the mix!
To find the absolute maximum and absolute minimum of a function, we need to analyze the critical points and endpoints of the given interval.
Given that the function is defined as f(x) = (x^3 - 4x^2 + 7x) / x, we can simplify it as f(x) = x^2 - 4x + 7.
To find the critical points, we need to find the derivative of f(x) and set it equal to zero:
f'(x) = 2x - 4
Setting f'(x) = 0, we get:
2x - 4 = 0
2x = 4
x = 2
Therefore, x = 2 is a critical point.
Next, let's determine the value of f(x) at the endpoints of the interval (0, 3]:
For x = 0:
f(0) = 0^2 - 4(0) + 7 = 7
For x = 3:
f(3) = 3^2 - 4(3) + 7 = 2
So, the values of f(x) at the endpoints are f(0) = 7 and f(3) = 2.
Now, we need to evaluate the value of f(x) at the critical point x = 2:
f(2) = 2^2 - 4(2) + 7 = -1
Finally, we compare the values obtained:
f(0) = 7 (endpoint)
f(2) = -1 (critical point)
f(3) = 2 (endpoint)
The absolute maximum is the maximum value among f(0) = 7, f(2) = -1, and f(3) = 2.
Therefore, the absolute maximum value is 7, which occurs at x = 0.
The absolute minimum is the minimum value among f(0) = 7, f(2) = -1, and f(3) = 2.
Therefore, the absolute minimum value is -1, which occurs at x = 2.
In conclusion, the absolute maximum is (0, 7), and the absolute minimum is (2, -1).
To find the absolute maximum and absolute minimum of a function on a closed interval, we need to evaluate the function at its critical points and endpoints.
First, let's find the critical points by taking the derivative of the function:
f(x) = (x^3 - 4x^2 + 7x) / x
f'(x) = (3x^2 - 8x + 7) / x
To find the critical points, we set f'(x) equal to zero and solve for x:
(3x^2 - 8x + 7) / x = 0
Multiplying both sides by x:
3x^2 - 8x + 7 = 0
Using the quadratic formula to solve for x:
x = (-(-8) ± √((-8)^2 - 4*3*7)) / (2*3)
x = (8 ± √(64 - 84)) / 6
x = (8 ± √(-20)) / 6
Since we have a negative value under the square root, there are no real solutions for x. Therefore, our function does not have any critical points in the interval (0, 3].
Next, we evaluate the function at the endpoints of the interval:
For x = 0:
f(0) = (0^3 - 4(0)^2 + 7(0)) / 0 = undefined
For x = 3:
f(3) = (3^3 - 4(3)^2 + 7(3)) / 3 = (27 - 36 + 21) / 3 = 12 / 3 = 4
So, the function is undefined at x = 0, and f(3) = 4.
Since the function is undefined at x = 0, we cannot consider it as a candidate for the absolute maximum or minimum. Therefore, the absolute maximum of f in the interval (0, 3] is (3, 4), and there is no absolute minimum in the given interval.
Based on the multiple-choice options you provided, (0, 7) is not a valid candidate for the absolute maximum of f in the given interval.