JEE Mathematics Application of Derivatives MCQs Set F

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: 120

Question:

• a) 10
• b) 12
• c) 11
• d) 9

Answer: 10

Question:

• a)

  

• b)

  

• c)

  

• d)

  

Answer:

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: R²/2

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: 4

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: P(–1) is not minimum but P(1) is the maximum of P

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: 1, 2 and 3 are correct

Question:

1) local maximum, if n is odd
(2) local maximum, if n is even
(3) local minimum, if n is even.
(4) local minimum, if n is odd

• a) 1 and 3 are correct
• b) 1 and 2 are correct
• c) 1, 2 and 3 are correct
• d) 2 and 4 are correct

Answer: 1 and 3 are correct

Question:

(1) minimum at x = 2 if n is even
(2) minimum at x = 1 if n is odd
(3) maximum at x = 1 if n is odd
(4) minimum at x = 1 if n is prime

• a) 1 and 3 are correct
• b) 1 and 2 are correct
• c) 1, 2 and 3 are correct
• d) 2 and 4 are correct

Answer: 1 and 3 are correct

Question: Let f (x) = e^(p+1) x – e^x for real number p > 0.

The value of x for which f (x) is minimum, is

• a)

  

• b) – ln (p + 1)
• c) – ln p
• d) None of these

Answer:

Question: Let f (x) = e^(p+1) x – e^x for real number p > 0.

• a)

  

• b)

  

• c)

  

• d) None of these

Answer:

Question: Let f (x) = e^(p+1) x – e^x for real number p > 0.

• a) 1/2
• b) does not exists
• c) 0
• d) 1

Answer: 1/2

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: Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1.

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: Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1.

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: Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1.

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: [0,2]

Question:

• a) Decreasing
• b) Increasing
• c) Neither increasing nor decreasing
• d) Increasing for x > 0 and decreasing for x < 0

Answer: Decreasing

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 ≤ -3

Question:

• a) Increases in [0,∞)
• b) Decreases in [0,∞)
• c) Neither increases nor decreases in [0,∞)
• d) Decreases in (-∞,∞)

Answer: Increases in [0,∞)

Question:

• a) f(x) is an increasing function
• b) g(x) is an increasing function
• c) Both f(x) and g(x) are decreasing functions
• d) Both f(x) and g(x) are increasing functions

Answer: f(x) is an increasing function

Question: The function which is neither decreasing nor increasing in

• a) cosec x
• b) x²
• c) tan x
• d) None of these

Answer: cosec x

Question: The function f (x) = sin⁴ x + cos⁴ x increases if :

• a)

  

• b)

  

• c)

  

• d)

  

Answer:

Question:

• a) –2
• b) 0
• c) –1
• d) –3

Answer: –2

Question:

• a)

  

• b)

  

• c)

  

• d)

  

Answer:

Question:

• a) 0
• b)

  

• c)

  

• d) None of these

Answer: 0

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: 0 < x <3/2

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: √ab

Question: The interval in which the function f(x) 4x²+1/x is decreasing is

• a)

  

• b)

  

• c) (–1, 1)
• d) [–1, 1]

Answer:

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: x < 0

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: increases

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: b > 1

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: increasing in ( – 2, 0) and in (1, ∞)

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: increasing in [–1/2, 1]

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: 3

Question:

• a) log_e (1 + x) < x
• b) log_e x > x
• c) e^x < 1 + x
• d) sin x > x

Answer: log_e (1 + x) < x

Question:

• a) 1 and 2 are correct
• b) 1 and 3 are correct
• c) 1, 2 and 3 are correct
• d) 2 and 4 are correct

Answer: 1 and 2 are correct

Question:

• a) 1 and 2 are correct
• b) 1 and 3 are correct
• c) 1, 2 and 3 are correct
• d) 2 and 4 are correct

Answer: 1 and 2 are correct

Question:

(1) 1 (2) 2
(3) 3 (4) 4

• a) 2 and 4 are correct
• b) 1, 2 and 3 are correct
• c) 1 and 2 are correct
• d) 1 and 3 are correct

Answer: 2 and 4 are correct

Question:

• a) Statement -1 is False, Statement-2 is True.
• b) Statement -1 is True, Statement-2 is False
• 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: Statement -1 is False, Statement-2 is True.

Question:

Statement 1 : If g (x) is a differentiable function. g (1) ≠ 0, g (–1) ≠ 0 and Rolles theorem is not applicable to f (x) = x²-1/g(x) in [–1,1], then g (x) has atleast one root in (–1, 1).

• a) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1.
• b) Statement-1 is True, Statement-2 is True; Statement-2 is 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: Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1.

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: Statement -1 is True, Statement-2 is False