Find the inflection points
y=x^2/x^+3
So far I've found y'=6x/(x^2+3)^2 but I'm stuck in finding y''. So far, I'm at 6(x^2+3)^2-24(x^2+3)/(x^2+3)^4 and the next step says y''=6(-3x^2+3)/(x^2+3)^3=0 and I don't understand how.
Thanks
looking at your answer for y' I will assume you meant:
y = x^2/(x^2 + 3)
your first derivative is correct, I suspect you used the quotient rule.
So why not use it also to get y''
y'' = ( (x^2 + 3)^2 (6) - 6x(2)(x^2 + 3) (2x) )/(x^2 + 3)^4
= ( 6(x^2+3)^2 - 24x^2 (x^2+3) )/(x^2 + 3)^4
= ( 6(x^2+3) (x^2 + 3 - 4x^2)/(x^2+3)^4
= 6(3 - 3x^2)/(x^2+3)^4
= 6(3-3x^2)/(x^2+3)^3
Your error was in the line I printed in bold.
assuming you mean
y = x^2/(x^2+3)
then
y' = 6x/(x^2+3)^2 is correct
y" = [6(x^2+3)^2 - 6x 2(x^2+3)2x ]/(x^2+3)^4
= [6(x^2+3) -24 x^2 ]/(x^2+3)^3
there is your error 6*x*2*2*x = 24 x^2 not 24
= 6 x^2 + 18 -24 x^2 is numerator
-18 x^2 + 18 = 0 for inflection point
x = +/-1
To find the inflection points of the function y = x^2 / (x^2 + 3), we need to find the second derivative and set it equal to zero.
Let's start with the first derivative:
y' = 6x / (x^2 + 3)^2
Now, to find the second derivative, we need to differentiate y' with respect to x. To simplify the process, let's use the quotient rule:
Quotient rule: (u/v)' = (u'v - uv') / v^2
Let u = 6x and v = (x^2 + 3)^2
Differentiating u with respect to x gives us u' = 6
Now, we can differentiate v using the chain rule:
Chain rule: (f(g(x)))' = f'(g(x)) * g'(x)
Let f = x^2 + 3 and g = x^2 + 3.
Differentiating f with respect to g gives us f' = 2(x^2 + 3)
Differentiating g with respect to x gives us g' = 2x
Now, using the quotient rule:
y" = [(u'v - uv') / v^2]' = [(6 * (x^2 + 3)^2 - 6x * 2(x^2 + 3) * 2x) / (x^2 + 3)^4]
Expanding and simplifying:
y" = [6(x^2 + 3)^2 - 24x^2(x^2 + 3)] / (x^2 + 3)^4
To find where the second derivative equals zero, you set the numerator equal to zero:
6(x^2 + 3)^2 - 24x^2(x^2 + 3) = 0
Now, let's factor out common terms:
6(x^2 + 3)^2 - 24x^2(x^2 + 3) = 0
6(x^2 + 3)[(x^2 + 3) - 4x^2] = 0
6(x^2 + 3)(3 - 3x^2) = 0
Now, we set each factor equal to zero and solve for x:
x^2 + 3 = 0 => x^2 = -3 => No real solutions since the square of any real number is always non-negative.
3 - 3x^2 = 0 => 3x^2 = 3 => x^2 = 1 => x = ±1
Hence, the second derivative equals zero at x = ±1.
These x-values represent the potential inflection points of the function y = x^2 / (x^2 + 3). To confirm if they are indeed inflection points, you can check the behavior of the function around those points by evaluating the concavity.