Question

For the function y = x²e^(-2x), at what values of x does the graph have a point of inflection?

a) x = 0
b) x = 2
c) x = 1
d) x = -2

Answer 1

To find the point of inflection for the function y = x²e^(-2x), one must identify when the second derivative of the function is equal to zero. By setting the second derivative equal to zero and simplifying the resulting quadratic equation, we find that the graph has a point of inflection when x = 1.

Step-by-step explanation:

To find the values of x where the graph of the function y = x²e^(-2x) has a point of inflection, we need to determine where the concavity of the graph changes. This involves finding the second derivative of the function and then solving for x when the second derivative is equal to zero.

Step-by-Step Solution

1. First, we find the first derivative of y:
   y' = d/dx (x²e^(-2x))
   Using the product rule and chain rule, we get:
   y' = 2xe^(-2x) - 2x²e^(-2x)
2. Next, we find the second derivative of y:
   y'' = d²/dx² (x²e^(-2x))
   Again using the product rule and chain rule, the result is:
   y'' = 2e^(-2x) - 4xe^(-2x) + 4x²e^(-2x)
3. To find the points of inflection, we set the second derivative equal to zero:
   0 = 2e^(-2x) - 4xe^(-2x) + 4x²e^(-2x)
   Factoring out e^(-2x), the equation simplifies to:
   0 = e^(-2x)(2 - 4x + 4x²).
4. Since e^(-2x) is never zero, we can focus on the quadratic equation:0 = 2 - 4x + 4x²
   We solve for x using the quadratic formula or factoring, yielding:x = 1.
5. Therefore, the graph has a point of inflection at x = 1 (option c).