For what values of x is the graph of y=x e^{-4 x} concave down?
y = xe^(-4x)
y' = (1-4x)e^(-4x)
y" = 8(2x-1)e^(-4x)
since e^(-4x) > 0 for all x, y" < 0 when 2x-1 < 0.
see the graph at
http://www.wolframalpha.com/input/?i=+xe^%28-4x%29+for+0+%3C%3D+x+%3C%3D+1
To determine the concavity of the graph of the function \(y = xe^{-4x}\), we need to find the second derivative and analyze its sign.
Step 1: Find the first derivative:
The first derivative of \(y = xe^{-4x}\) with respect to \(x\) can be found using the product rule:
\(y' = (x)'e^{-4x} + x(e^{-4x})'\)
Using the power rule and chain rule, we get:
\(y' = (1)(e^{-4x}) + x(-4e^{-4x})\)
Simplifying further:
\(y' = e^{-4x} - 4xe^{-4x}\)
Step 2: Find the second derivative:
To find the second derivative, we differentiate the first derivative with respect to \(x\):
\(y'' = (e^{-4x} - 4xe^{-4x})'\)
Applying the product rule again:
\(y'' = (e^{-4x})' - (4xe^{-4x})'\)
Differentiating each term separately using the chain rule:
\(y'' = (-4e^{-4x}) - (4e^{-4x} - 4x(-4e^{-4x}))\)
Simplifying:
\(y'' = -4e^{-4x} - 4e^{-4x} + 16xe^{-4x}\)
Combining like terms:
\(y'' = -8e^{-4x} + 16xe^{-4x}\)
Step 3: Analyze the sign of the second derivative:
To determine when the graph is concave down, we need to analyze the sign of the second derivative \(y''\).
The second derivative is \(y'' = -8e^{-4x} + 16xe^{-4x}\).
For the graph to be concave down, \(y''\) should be negative.
Setting \(y'' < 0\), we have:
\(-8e^{-4x} + 16xe^{-4x} < 0\)
Using the common factor \(e^{-4x}\):
\(e^{-4x}(-8 + 16x) < 0\)
Since \(e^{-4x}\) is always positive, we can ignore it.
Now we need to analyze the sign of the expression \(-8 + 16x\).
Setting \(-8 + 16x < 0\):
\(16x < 8\)
Dividing both sides by 16:
\(x < \frac{1}{2}\)
Therefore, the graph of \(y = xe^{-4x}\) is concave down for \(x < \frac{1}{2}\).
Note: This does not include the boundary point \(x = \frac{1}{2}\). To determine the concavity at that point, we can substitute \(x=\frac{1}{2}\) into the second derivative and analyze the sign.
To determine the concavity of the graph of the function y = x e^(-4x), we need to find the second derivative of the function and analyze its behavior.
Step 1: Find the first derivative of the function y = x e^(-4x).
Taking the first derivative will give us the slope of the tangent line to the graph at any given point.
dy/dx = d/dx (x e^(-4x)) = e^(-4x) + (-4x)e^(-4x) = (1-4x)e^(-4x)
Step 2: Find the second derivative of the function.
Taking the derivative of the first derivative will give us the rate of change of the slope or the concavity of the graph.
d^2y/dx^2 = d/dx ((1-4x)e^(-4x)) = (d/dx (1-4x))e^(-4x) + (1-4x)d/dx (e^(-4x))
= (-4)e^(-4x) + (1-4x)(-4)e^(-4x) = (-4 - 4(1-4x))e^(-4x)
= (-4 + 16x - 4x)e^(-4x) = (12x - 4)e^(-4x)
Step 3: Determine when the second derivative is negative.
To find when the graph is concave down, we need the second derivative to be negative.
(12x - 4)e^(-4x) < 0
Step 4: Solve the inequality to find the values of x.
Since e^(-4x) is always positive, we can ignore it when determining the concavity.
12x - 4 < 0
12x < 4
x < 4/12
x < 1/3
Therefore, the graph of y = x e^(-4x) is concave down for values of x less than 1/3.