Find the x-coordinates of any relative extrema and inflection point(s) for the function f(x) = 3x^(1/3) + 6x^(4/3). Please use an analysis of f ′(x) and f ′′(x).
if y = 3x^(1/3) + 6x^(4/3)
then
dy/dx = x^-(2/3)+ 8 x^(1/3)
and
d^2y/dx^2
=-(2/3)x^-(5/3)+(8/3)x^-(2/3)
extrema or inflection where dy/dx = 0
note slope undefined when x = 0
dy/dx = x^-(2/3)+ 8 x^(1/3) = 0
or
1/x^(2/3) = - 8 x^(1/3)
1 = -8 x
x = -1/8
so what is d^2/dy^2 when x = -1/8 ?
d^2y/dx^2
=-(2/3)x^-(5/3)+(8/3)x^-(2/3)
= -(2/3)(-1/8)^-5/3 + (8/3) (-1/8)^-(2/3)
well -1/8^(5/3) = (-1/2)^5 = -1/32
so
(-1/8)^-5/3 = -32
and we have so far
-(2/3)(-32) +(8/3) (-1/8)^-(2/3)
64/3 + (-1/2)^-2 = 1/(1/4) = 4
64/3 + 4
so second derivative is positive, a relative minimum at x = -1/8
CHECK MY ARITHMETIC !!!
You got it right bro
To find the x-coordinates of any relative extrema and inflection point(s) for the function f(x) = 3x^(1/3) + 6x^(4/3), we need to analyze its first derivative, f'(x), and second derivative, f''(x).
First, let's find the first derivative f'(x). To do this, we differentiate each term of the function:
f'(x) = d/dx (3x^(1/3)) + d/dx (6x^(4/3))
To differentiate x^(n), we use the power rule, which states that d/dx (x^n) = n * x^(n-1).
So applying the power rule, we have:
f'(x) = 3 * d/dx (x^(1/3)) + 6 * d/dx (x^(4/3))
= 3 * (1/3) * x^(-2/3) + 6 * (4/3) * x^(1/3)
= x^(-2/3) + 8x^(1/3)
Now, let's find the second derivative f''(x) by differentiating f'(x):
f''(x) = d/dx (x^(-2/3) + 8x^(1/3))
= d/dx (x^(-2/3)) + d/dx (8x^(1/3))
= (-2/3) * x^(-5/3) + (8/3) * x^(-2/3)
Now that we have the first and second derivatives, we can use them to find the relative extrema and inflection point(s).
1. Relative Extrema:
To find the relative extrema, we need to solve the equation f'(x) = 0. Set f'(x) = 0 and solve for x:
x^(-2/3) + 8x^(1/3) = 0
We can multiply both sides by x^(2/3) to simplify the equation:
1 + 8x = 0
Now, solve for x:
8x = -1
x = -1/8
Thus, the x-coordinate of the relative extremum is x = -1/8.
2. Inflection Point(s):
To find the inflection point(s), we need to solve the equation f''(x) = 0. Set f''(x) = 0 and solve for x:
(-2/3) * x^(-5/3) + (8/3) * x^(-2/3) = 0
Since the two terms have different powers of x, it is not easy to solve algebraically. We can use numerical methods or graphical methods to estimate the inflection points.
By analyzing the signs of f''(x) on either side of the point where f''(x) changes sign, we can determine the presence of an inflection point(s).
Hence, the x-coordinates of the relative extrema are x = -1/8, and the x-coordinates of the inflection point(s) can be determined using further analysis.