48. How many critical values does the function f(x) = arctan(2x − x²) have?






Answer is: option2

1

Solution:

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