Gaokao Exam Gaokao Mathematics: Functions and Calculus 6

Question 1

If f(x) = sin(x) - cos(x), the maximum value of f(x) is:

Accepted Answer

√2

Answer

f(x) = √2 sin(x - π/4); amplitude = √2, so maximum value is √2.

Question 2

Find the inflection point of f(x) = x³ - 3x² + x + 1.

Accepted Answer

x = 1

Answer

f''(x) = 6x - 6 = 0 → x = 1; f''changes sign at x=1 (negative to positive), so x=1 is an inflection point.

Question 3

Which of the following pairs (f, g) satisfies f(x) + g(x) = 0 for all x?

Accepted Answer

f(x) = x², g(x) = -x²

Answer

x² + (-x²) = 0 for all x; the other pairs do not sum to zero identically.

Question 4

Using L'Hôpital's rule, evaluate lim_{x→0} (sin x)/x.

Accepted Answer

1

Answer

0/0 form; differentiating numerator and denominator: lim (cos x)/1 = cos(0) = 1.

Question 5

f(x) = x·|x|. Which statement is true?

Accepted Answer

f is differentiable everywhere with f'(0) = 0

Answer

For x > 0: f(x) = x², f'(x) = 2x; for x < 0: f(x) = -x², f'(x) = -2x; at x=0: both one-sided limits → 0; f'(0) = 0. So f is differentiable everywhere.

Gaokao Exam Practice Test

Gaokao Exam Practice Test

Gaokao Exam Gaokao Mathematics: Functions and Calculus 6