Find the interval(s) where the function is increasing of decreasing.
find the:
a) critical value(s)
b) critical point(s)
c) max. value + max. point
d) min.value and min. point
e)point on inflection if there is:
1) y=x^4-3x^3+22x^2-24x+12
thanks:)
A rather uninspiring graph
There is an obvious y-intercept of 12
after trying the usual methods of finding some f(a) = 0
I used WolframAlpha to see what we had
http://www.wolframalpha.com/input/?i=x%5E4-3x%5E3%2B22x%5E2-24x%2B12+
It showed no x-intercepts, thus no solution for f(x) = 0 , but rather 4 complex roots
I then clicked on the derivative expression to get one real root at appr x = .6
f(.6) = .6^4 - 3(.6)^3 + 22(.6)^2 - 24(.6) + 12 = appr. 5
So there is a turning point at about (0.6 , 5)
see http://www.wolframalpha.com/input/?i=-24%2B44+x-9+x%5E2%2B4+x%5E3&lk=1
clicking on the derivative of that cubic shows that the resulting quadratic has no real roots, thus no points of inflection
(It would have been a nightmare to attempt this without some convenient software, solving this with only pencil and paper and a basic calculator seems daunting )
To find the intervals where the function is increasing or decreasing, you need to analyze the first derivative of the function.
Step 1: Find the first derivative of the function:
f'(x) = 4x^3 - 9x^2 + 44x - 24
Step 2: Find the critical values:
To find the critical values, set the first derivative equal to zero and solve for x.
4x^3 - 9x^2 + 44x - 24 = 0
You can either use factoring or numerical methods like Newton's Method or the Newton-Raphson method to solve this equation. Once you find the values of x, these are the critical values.
Step 3: Find the intervals where the function is increasing or decreasing:
To determine the intervals of increasing or decreasing, you can use the value of the first derivative on either side of the critical values:
- Choose a value to the left (prior) of the smallest critical value, plug it into the first derivative (f'(x)).
- Choose a value between each pair of critical values and do the same.
- Choose a value to the right (later) of the largest critical value and repeat the process.
If f'(x) > 0, the function is increasing in that interval.
If f'(x) < 0, the function is decreasing in that interval.
Step 4: Find the critical points:
The critical points are the x-values where the function may have local maxima or minima. To find these points, substitute the critical values into the original function and find the corresponding y-values.
Step 5: Find the maximum and minimum values and points:
Once you have the critical points, substitute them into the original function to get the corresponding y-values. The maximum value would be the highest y-value among these critical points, and the minimum value would be the lowest y-value.
Step 6: Find the points of inflection (if any):
To find the points of inflection, you need to analyze the concavity of the function using the second derivative.
Step 7: Find the second derivative:
f''(x) = 12x^2 - 18x + 44
Step 8: Find the values of x where f''(x) = 0:
Set f''(x) equal to zero and solve for x.
12x^2 - 18x + 44 = 0
Again, you can use factoring or numerical methods to solve this equation. The resulting x-values would be the potential points of inflection.
Step 9: Confirm the points of inflection:
To determine whether these x-values are indeed points of inflection, analyze the behavior of the second derivative (f''(x)) on either side of these values. If the concavity changes from positive to negative or vice versa, then these points are confirmed as points of inflection.
By following these steps, you should be able to find the intervals of increasing or decreasing, the critical values, critical points, maximum value and point, minimum value and point, and any points of inflection for the given function y = x^4 - 3x^3 + 22x^2 - 24x + 12.