AP ECET MATHEMATICS Multiple Choice Questions (MCQs)

91. If f'(x) > 0 for all x in an interval, then f(x) is:

A) Increasing in that interval

B) Decreasing in that interval

C) Constant in that interval

D) None of the above

Answer: A) Increasing in that interval

92. If f'(x) < 0 for all x in an interval, then f(x) is:

A) Increasing in that interval

B) Decreasing in that interval

C) Constant in that interval

D) None of the above

Answer: B) Decreasing in that interval

93. The function f(x) = x^3 - 3x + 2 has a local maximum at:

A) x = 1

B) x = -1

C) x = 0

D) No local maximum exists

Answer: A) x = 1

94. To determine the nature of a critical point, which of the following is used?

A) First derivative test

B) Second derivative test

C) Both A and B

D) None of the above

Answer: B) Second derivative test

95. If f''(x) > 0 at a point, then the point is:

A) A local minimum

B) A local maximum

C) An inflection point

D) A critical point

Answer: A) A local minimum

96. The function f(x) = 4x^3 - 6x^2 + 2x has a local minimum at:

A) x = 1

B) x = 0

C) x = -1

D) No local minimum exists

Answer: A) x = 1

97. The rate of change of a quantity with respect to time is an example of:

A) Differentiation as rate measure

B) Maxima and minima

C) Partial differentiation

D) Error approximation

Answer: A) Differentiation as rate measure

98. The formula for the linear approximation of a function f(x) around x = a is:

A) f(x) = f(a) + f'(a)(x - a)

B) f(x) = f(a) - f'(a)(x - a)

C) f(x) = f'(a) + f(a)(x - a)

D) None of the above

Answer: A) f(x) = f(a) + f'(a)(x - a)

99. The second-order partial derivative of f(x, y) = x^2 y + y^3 with respect to x is:

A) 2y

B) 2x

C) 2xy

D) 3y^2

Answer: B) 2x

100. Euler's Theorem relates to:

A) Homogeneous functions

B) Implicit functions

C) Derivatives of parametric functions

D) Errors and approximations

Answer: A) Homogeneous functions

101. The second-order partial derivative of f(x, y) = x^3 y^2 with respect to y is:

A) 3x^3 y

B) 2x^3 y

C) 6x^3 y

D) 6x^3 y^2

Answer: C) 6x^3 y

102. If f(x) is a function such that f'(x) = 0 and f''(x) > 0, then the function has:

A) A local maximum at x

B) A local minimum at x

C) An inflection point at x

D) No critical point

Answer: B) A local minimum at x

103. The first-order partial derivative of the function f(x, y) = x^2 + y^2 with respect to x is:

A) 2x

B) 2y

C) 2x + 2y

D) x + y

Answer: A) 2x

104. In the second derivative test, if f''(x) = 0 at a critical point, we can conclude:

A) The point is an inflection point

B) The point is a local minimum

C) The point is a local maximum

D) Further analysis is required

Answer: D) Further analysis is required

105. The error in the approximation of a function can be minimized by:

A) Using a higher-order derivative in the Taylor series

B) Using first-order derivative only

C) Using only second-order derivatives

D) Using partial derivatives

Answer: A) Using a higher-order derivative in the Taylor series