To find the maximum value of the function f(x) = x2e2x for x ≥ 0, we first need to take its derivative and find the critical points.
1. **Derivative Calculation**: Using the product rule, we differentiate f(x):
Let u = x2 and v = e2x. Then, the derivatives are u’ = 2x and v’ = 2e2x. Applying the product rule:
f'(x) = u’v + uv’ = 2xe2x + x2(2e2x) = e2x(2x + 2x2) = 2xe2x(x + 1).
2. **Finding Critical Points**: Set the derivative equal to zero:
2xe2x(x + 1) = 0.
This gives us two cases:
- 2x = 0 which implies x = 0
- x + 1 = 0 which gives a negative solution and is not considered since x ≥ 0.
Thus, the only critical point is x = 0.
3. **Evaluate the Function**: Now we need to evaluate f(0):
f(0) = 02e0 = 0.
4. **Check Behavior as x Approaches Infinity**: As x increases towards infinity, e2x grows very rapidly and dominates the x2 term:
Thus, as x → ∞, f(x) → ∞. Therefore, there is no maximum value at a finite point.
In conclusion, the function does not have a maximum value at finite x, but trends towards infinity as x increases.