JEE Mathematics Application of Derivatives MCQs Set 06

Question. The function f(x) = x⁴ – 62x² + ax + 9 attains its maximum value on the interval [0, 2] at x = 1. Then the value of a is:
(a) 120
(b) 52
(c) –120
(d) None of these.

Answer: A

Question. The maximum area of rectangle inscribed in a circle of diameter R is:
(a) R²/2
(b) R²/8
(c) R²
(d) R²/4

Answer: A

Question. If sum of two numbers is 3, the maximum value of the product of first and the square of second is:
(a) 4
(b) 2
(c) 3
(d) 1

Answer: A

Question. Given P(x) = x⁴ + ax³ + bx² + cx + d such that x = 0 is the only real root of P¢(x) = 0. If P(–1) < P(1), then in the interval [ –1, 1] :
(a) P(–1) is not minimum but P(1) is the maximum of P
(b) P(–1) is the minimum but P(1) is not the maximum of P
(c) Neither P(–1) is the minimum nor P(1) is the maximum of P
(d) P(–1) is the minimum and P(1) is the maximum of P

Answer: A

Question. If a, b, c are natural numbers and ax⁴- bx³ +cx² -bx+a/ (x²+1)² attains minimum value at x=2 or x= 1/2  then the least possible values of a, b, c are respectively
(1) 1, 4, 7 (2) 1, 8, 12
(3) 2, 4, 9 (4) 1, 2, 3
(a) 1, 2 and 3 are correct
(b) 2 and 4 are correct
(c) 1 and 2 are correct
(d) 1 and 3 are correct

Answer: A

Question. Statement 1 : Among all the rectangles of given perimeter, the square has the largest area. Also among all the rectangles of given area, the square has the least perimeter.
Statement 2 : For x > 0, y > 0, if x + y = const, then xy will be maximum for y = x and if xy = const., then x + y will be minimum for y = x.
(a) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1.
(b) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1.
(c) Statement -1 is False, Statement-2 is True.
(d) Statement -1 is True, Statement-2 is False.

Answer: A

Question. Statement 1 : If f (x) = (x – 3)³, then f (x) has neither maximum nor minimum at x = 3
Statement 2 : f ' (x) = 0, f '' (x) = 0 at x = 3.
(a) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1.
(b) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1.
(c) Statement -1 is False, Statement-2 is True.
(d) Statement -1 is True, Statement-2 is False.

Answer: A

Question. Statement 1 : If f (x) = max {x² – 2x + 2, | x – 1 |}, then the greatest value of f (x) on the interval [0, 3] is 5.
Statement 2 : Greatest value of f (3) = max. {5, 2} = 5.
(a) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1.
(b) Statement -1 is False, Statement-2 is True.
(c) Statement -1 is True, Statement-2 is False.
(d) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1.

Answer: A

Question. The interval in which the function x²e^-x is non decreasing, is
(a) [0,2]
(b) (-∞,0]
(c) None of these
(d) [2,∞)

Answer: A

Question. The values of 'a ' for which the function (a + 2)x³ - 3ax² + 9ax –1 decreases monotonically throughout for all real x , are
(a) -∞ < a ≤ -3
(b) 33 < a < 0
(c) a < – 2
(d) a > – 2

Answer: A

Question. y = [x (x – 3)]² increases for all values of x lying in which of the following interval?
(a) 0 < x <3/2
(b) 0 < x < ∞
(c) 1 < x < 3
(d) None of these

Answer: A

Question. From Mean value theorem f(b) – f(a) = (b – a) f ' (x₁) where a < x1 < b and f(x) =1/x then x equal to:
(a) √ab
(b) 2ab/a +b
(c) b-a/b+a
(d) a+b/2

Answer: A

Question. If a < 0, the function f (x) = e^ax + e^–ax is a monotonically decreasing function for values of x given by :
(a) x < 0
(b) x < 1
(c) x > 0
(d) x > 1

Answer: A

Question. If the function f : R→R is defined by f (x) = tan x – x, then f '(x) is :
(a) increases
(b) constant
(c) decreases
(d) none of these

Answer: A

Question. The value of b for which the function f (x) = sin x – bx + c is decreasing in the interval (-∞,∞) is given by
(a) b > 1
(b) b < 1
(c) b ≥1
(d) b ≤ 1

Answer: C

Question. If f (x) = 3x⁴ + 4x³ – 12x² + 12, then f (x) is
(a) increasing in ( – 2, 0) and in (1, ∞)
(b) decreasing in ( – 2, 0) and in (0,1)
(c) decreasing in ( – ∞ , – 2) and in (1, ∞)
(d) increasing in (– ∞ , – 2) and in (0, 1)

Answer: A

Question. If f (x) = xe^x(1–x) , then f (x) is
(a) increasing in [–1/2, 1]
(b) increasing in R
(c) decreasing in R
(d) decreasing in [–1/2, 1]

Answer: A

Question. Let f (x) and g(x) be differentiable for 0 ≤ x ≤ 1, such that f( 0) = 0, g (0) = 0, f(1) = 6. Let there exist a real number c in (0, 1) such that f'(c) = 2 g'(c) , then the value of g (1) is
(a) 3
(b) 0
(c) –3
(d) None of these

Answer: A

Question. Statement 1 : If f (x) is increasing function with concavity upwards, then concavity of f^–1 (x) is also upwards.
Statement 2 : If f (x) is decreasing function with concavity upwards, then concavity of f^–1 (x) is also upwards.
(a) Statement -1 is True, Statement-2 is False
(b) Statement -1 is False, Statement-2 is True.
(c) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1.
(d) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1.

Answer: A