Mathematics, 31.01.2020 05:46, jmallory3031

Y=yx^3-3x+2 find the intervals where the function is concave up or down and inflection points

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$\bf y=yx^3-3x+2\impliedby \textit{using implicit differentiation}
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\cfrac{dy}{dx}=\cfrac{dy}{dx}x^3+y3x^2-3\implies \cfrac{dy}{dx}-\cfrac{dy}{dx}x^3=3yx^2-3
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\cfrac{dy}{dx}(1-x^3)=3yx^2-3\implies \boxed{\cfrac{dy}{dx}=\cfrac{3yx^2-3}{1-x^3}}\\\\
-----------------------------\\\\
\cfrac{dy}{dx}=\cfrac{3yx^2-3}{1-x^3}\impliedby \textit{using the quotient rule}
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\cfrac{dy^2}{dx^2}=\cfrac{3[2xy+x^2\frac{dy}{dx}(1-x^3)]-(3x^2y-3)(-3x^2)}{(1-x^3)^2}
\\\\\\
$

$\bf \begin{cases}
3[2xy+x^2\frac{dy}{dx}](1-x^3)\implies \left[6xy+6x^2\cdot \frac{3yx^2-3}{1-x^3}\right](1-x^3)\\
6xy-6x^4y+18x^4y-18x^2
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----------------\\
(3x^2y-3)(-3x^2)\implies -9x^4y+9x^2\\
\qquad -(9x^4y+9x^2)
\end{cases}
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\cfrac{dy^2}{dx^2}=\cfrac{6xy-6x^4y+18x^4y-18x^2+9x^4y+9x^2}{(1-x^3)^2}
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\boxed{\cfrac{dy^2}{dx^2}=\cfrac{6xy+21x^4y-9x^2}{(1-x^3)^2}}$

so hmmm since it's implicit differentiation, you'd get a function in x,y terms so. hmm if we check the graph of that second derivative, it seems to be asymptotic at x = -1, meaning an inflection point, from the denominator equaling 0

and it has an x-intercept at the origin, x = 0, another inflection point

now, the graph of the second derivative, from x > 0, the value for "y" is positive all the way, with a horizontal asymptote at y = 0
meaning the original function f(x) is concave UP at x > 0

now, when -1 < x < 0, the value of "y" drops to the negatives, thus is concave DOWN at -1 < x < 0

and if we pass the asymptotic inflection point of x = -1, to the left
x < -1, the value of "y" is on the positives, thus concave UP on that region

answered: Guest

b.) 160 miles on a 5 gallons car

step-by-step explanation:

240/10=24 miles/gallon

160/5=32 miles/gallon

30/1=30 miles/gallon

168/6= 28 miles/gallon

32 miles/gallon is the most amount, so this is the best.

answered: Guest

answer: ratio of volume of cone (with 3 times the height and equal diameter) to volume of cylinder is 2 to 1. the value of π is not used.

step-by-step explanation:

radius is diameter/2 = 4.