Answer is: option2
1Solution:
Compute the derivative f'(x):
The derivative of arctan(u) with respect to x is u'⁄1 + u², where u = 2x − x².
f'(x) = d/dx (arctan(2x − x²)) = (2 − 2x) / (1 + (2x − x²)²)
Find where f'(x) = 0 or is undefined:
- Numerator:
2 − 2x = 0
2x = 2
x = 1 - Denominator:
1 + (2x − x²)² is always positive (since (2x − x²)² ≥ 0), so the derivative is never undefined.
Therefore, the only critical point is at x = 1.
The function f(x) has 1 critical value.
Answer: B