find the relative extrema and points of inflection, if possible, of y = xln((x^4)/8)
Any help would be awesome...thanks!
(i know that the relative extrema come from critical values from y' and points of inflection come from y'')
dy/dx = y' = ln((x^4)/8) + x(4/x)
= ln ( (x^4)/8 ) + 4
y '' = 4/x
for relative extrema, y' = 0
ln ((x^4)/8) = 0
x^4 / 8 = e^0 = 1
x^4 = 8
x = ± 8^(1/4)
plug in those values into y = ...
for pts of inflection, 4/x = 0 ----> no solution
thus, no points of inflection
And yet, y'' changes from - to + as x increases through 0, so while y is discontinuous at x=0, there appears to be an inflection point there.
To get the limit of y at x=0,
y(0) = 0*(-oo)
setting t=1/x and using L'Hopital's rule a bit, we can see that the limit is 0.
To find the relative extrema and points of inflection of the function y = xln((x^4)/8), you'll need to follow a few steps.
Step 1: Find the derivative of the function y' = dy/dx.
Start by applying the product rule. Let's break down the function to make it easier:
y = x * ln((x^4)/8)
y = x * (ln(x^4) - ln(8))
y = 4x * ln(x) - x * ln(8)
Now, take the derivative of y with respect to x:
y' = 4x * d/dx(ln(x)) - ln(8)
Step 2: Simplify y'.
To simplify y', we need to find the derivative of ln(x), which is 1/x. Thus, we have:
y' = 4x * (1/x) - ln(8)
y' = 4 - ln(8)
Step 3: Set y' equal to zero to find critical points.
To find the critical points, set y' = 4 - ln(8) equal to zero and solve for x:
4 - ln(8) = 0
ln(8) = 4
e^ln(8) = e^4
8 = e^4
Therefore, x = 8 is the critical point.
Step 4: Determine the type of extrema.
To determine the type of extrema (maximum or minimum), we can use the second derivative test. However, since y' is constant, there are no relative extrema for this function.
Step 5: Find points of inflection.
To find the points of inflection, we need to determine the second derivative, y'' = d^2y/dx^2:
y'' = d/dx(4 - ln(8))
y'' = 0 - 0
y'' = 0
Since y'' equals zero, there are no points of inflection for this function.
In summary, y = xln((x^4)/8) has no relative extrema but also no points of inflection. The only critical point is x = 8.