x(x+4)^(2/3)
a). Find the second derivative
b). Find any points of Inflection
c). Determine the intervals of Concavity
y = x(x+4)^(2/3)
y' = (x+4)^(2/3) + 2x/3 * (x+4)^(-1/3)
= (5x+12)/(3(x+4)^(1/3))
y'' = 2(5x+24)/(9(x+4)^(4/3))
Since the denominator is always positive, we just need to look at the numerator.
inflection points where f '' = 0, x=-24/5
That's where it changes from concave down to concave up.
However, watch out for x = -4, where y' and y'' are undefined.
1st deriv
= x(2/3)(x+4)^(-1/3) + (x+4)^(2/3)
= (1/3)(x+4)^(-1/3) [ 2x+ 3(x+4) ]
= (1/3)(5x+12)(x+4)^(-1/3)
2nd deriv
= (1/3)(5x+12)(-1/3)(x+4)^(-4/3) + (1/3)(5))x+4)^(-1/3)
= ....
= 2(5x+24)/(9(x+4)^(4/3) )
should be easy from there
To find the second derivative, we need to differentiate the function twice. Let's start with the first derivative.
a) First Derivative:
To find the first derivative of the function f(x) = x(x+4)^(2/3), we can use the product rule. The product rule states that if we have two functions, u(x) and v(x), their derivative is given by (u'v + uv').
Let's begin by differentiating u(x) = x, and v(x) = (x+4)^(2/3).
1. Differentiating u(x):
Since u(x) = x, the derivative of x with respect to x is simply 1.
2. Differentiating v(x):
To differentiate v(x) = (x+4)^(2/3), we need to use the chain rule. The chain rule states that if we have a composite function, u(v(x)), its derivative is given by (u'(v) * v').
Let u(v) = v^(2/3). Then, differentiating u(v) with respect to v using the power rule, we get u'(v) = (2/3)v^(-1/3).
Now, we differentiate v(x) = x+4 with respect to x to get v'(x) = 1.
3. Applying the product rule:
Using the product rule, we can now find the first derivative of f(x) = x(x+4)^(2/3):
f'(x) = u'v + uv'
= (1)(v) + (x)((2/3)(x+4)^(-1/3))
Simplifying this expression gives us the first derivative: f'(x) = (x+4)^(2/3) + (2/3)x(x+4)^(-1/3).
b) Points of Inflection:
To find the points of inflection, we need to locate the x-values where the concavity of the function changes. These occur when the second derivative is zero or undefined.
Since we have not calculated the second derivative yet, we'll proceed to find it.
2. Second Derivative:
To find the second derivative, we differentiate the first derivative we obtained earlier, f'(x).
Differentiating f'(x) = (x+4)^(2/3) + (2/3)x(x+4)^(-1/3), we need to apply the product rule again. However, before that,
1. Differentiating (x+4)^(2/3):
To differentiate (x+4)^(2/3), we use the chain rule where u(v) = v^(2/3).
Differentiating u(v) = v^(2/3) with respect to v using the power rule, we get u'(v) = (2/3)v^(-1/3).
2. Differentiating (2/3)x(x+4)^(-1/3):
To differentiate (2/3)x(x+4)^(-1/3), we use the product rule.
Differentiating u(x) = (2/3)x with respect to x gives u'(x) = 2/3.
Differentiating v(x) = (x+4)^(-1/3) with respect to x requires the chain rule.
Let u(v) = v^(-1/3). Then, differentiating u(v) with respect to v using the power rule, we get u'(v) = (-1/3)v^(-4/3).
Now, differentiating v(x) = (x+4)^(-1/3) with respect to x using the chain rule, we obtain v'(x) = (-1/3)(x+4)^(-4/3).
3. Applying the product rule:
Using the product rule, we can now find the second derivative of f(x):
f''(x) = u'v + uv'
= (2/3) (x+4)^(-1/3) + (x)((-1/3)(x+4)^(-4/3))
Simplifying this expression gives us the second derivative: f''(x) = (2/3)(x+4)^(-1/3) - (1/3)x(x+4)^(-4/3).
c) Intervals of Concavity:
To determine the intervals of concavity, we need to find the x-values where the second derivative is positive or negative. Positive values indicate concavity upwards (concave up), while negative values indicate concavity downwards (concave down).
To do this, we can examine the signs of the second derivative at different intervals. To find these intervals, we need to find where the second derivative is equal to zero or undefined and any other critical points.
Now that we have the second derivative, f''(x) = (2/3)(x+4)^(-1/3) - (1/3)x(x+4)^(-4/3), we can analyze its sign to determine intervals of concavity and points of inflection.