Consider the function f(x)=12x^5+30x^4–300x^3+2. f(x) has inflection points at (reading from left to right) x=D,E, and F. where D is___, E is____, and F is____.
I took the second derivative of the function and got 120x(2x^2+3x-15). From that I set it to equal and tried finding x. One is obviously zero.
Then I tried using the quadratic formula to get the other two x's. I got [-3 + sqrt(129)]/4 and [-3 - sqrt(129)]/4
However I plugged the answer into my online homework site and it says that is incorrect. Can someone help?
Assistance needed.
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I agree with you.
Writeacher, thanks for the tip...I was never really sure whether I was supposed to type my actual class subject or the subject material. If it's ok, I might submit this question again with a different school subject.
Finding the inflection points of a function involves locating the values of x where the concavity of the function changes. To do this, you correctly took the second derivative of the function f(x).
Let's simplify the second derivative expression you obtained: 120x(2x^2 + 3x - 15).
To find the x-values of the inflection points, we need to set this expression equal to zero.
120x(2x^2 + 3x - 15) = 0
Now, we solve for x. Since 120x is never zero (unless x equals zero), we have two cases to consider:
Case 1: 2x^2 + 3x - 15 = 0
For this quadratic equation, we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = 2, b = 3, and c = -15. Substituting these values into the quadratic formula, we get:
x = (-3 ± √(3^2 - 4(2)(-15))) / (2(2))
x = (-3 ± √(9 + 120)) / 4
x = (-3 ± √129) / 4
So, the two possible values for x in this case are:
D ≈ (-3 + √129) / 4
E ≈ (-3 - √129) / 4
Case 2: 120x = 0
From this equation, we find one solution:
F = 0
Therefore, the inflection points of f(x) are approximately:
D ≈ (-3 + √129) / 4
E ≈ (-3 - √129) / 4
F = 0
It is worth mentioning that the provided values of D, E, and F are approximations since √129 is an irrational number. Make sure to double-check if the provided answer format matches your online homework site's required format.