For the function g(x) = xe power x, there is inflection point at?
g(x) = xe^x
g'(x) = xe^x + e^x
g''(x) = xe^x + e^x + e^x
= e^x( x+2)
at point of inflection g''(x) = 0
so e^x = 0 ----> no solution
or x+2 = 0
x = -2
then g(-2) = -2e^-2 or -2/e^2
point of inflection (-2, -2/e^2)
The average value of f(x) = e ^ 4xsquared on the interval [-1/4,1/4]
To find the inflection point of a function, we need to determine the values of x where the curvature changes.
Step 1: Find the first and second derivatives of the function g(x).
The derivative of g(x) with respect to x is obtained by applying the product rule.
g'(x) = (x * e^x)' = (1 * e^x) + (x * e^x) = e^x + x * e^x = (1 + x) * e^x.
The second derivative of g(x) with respect to x can be found by differentiating g'(x) with respect to x.
g''(x) = [(1 + x) * e^x]' = (e^x + xe^x)'
= (e^x)' + (xe^x)'
= e^x + e^x + xe^x = 2e^x + xe^x = (2 + x)*e^x.
Step 2: Determine the values of x that make the second derivative, g''(x), equal to zero.
Setting g''(x) = 0, we have:
(2 + x)*e^x = 0.
Since e^x is always positive, we can conclude that (2 + x) must equal zero, that is:
2 + x = 0.
Solving for x, we find:
x = -2.
Therefore, the inflection point for the function g(x) = x * e^x is located at x = -2.