The total number of local maxima and local minima of the function f (x) = {(2 + x)3, when – 3 < x ≤ -1 and x2/3, when -1 < x < 2 is 1) 0 2) 1 3) 2 4) 3 Solution: (3) 2 f (x) = {(2 + x)3, when - 3 < x ≤ -1 and x2/3, when -1 < x < 2 f’ (x) = f (x) = {3... View Article
If from a wire of length 36 metre a rectangle of greatest area is made, then its two adjacent sides in metre are 1) 6, 12 2) 9, 9 3) 10, 8 4) 13, 5 Solution: (2) 9, 9 2 (a + b) = 36 a + b = 18 The area of the rectangle is ab = a (18 - a) A =... View Article
Maximum value of x (1 – x)2 when 0 ≤ x ≤ 2, is 1) 2 / 27 2) 4 / 27 3) 5 4) 0 Solution: (2) 4 / 27 f (x) = x (1 - x)2 f (x) = x3 - 2x2 + x f' (x) = 3x2 - 4x + 1 f'(x) = 0,... View Article
The minimum value of f (a) = (2a2 – 3) + 3 (3 – a) + 4 is 1) 15 / 2 2) 11 / 2 3) -13 / 2 4) 71 / 8 Solution: (4) 71 / 8 f (a) = 2a2 - 3a + 10 f’ (a) = 4a - 3, f’ (a) = 4 > 0 f (a) is... View Article
If the function f (x) = x4 – 62x2 + ax + 9 is maximum at x = 1, then the value of a is 1) 120 2) –120 3) 52 4) 128 Solution: (1) 120 f (x) = x4 - 62x2 + ax + 9 f′ (x) = 4x3 − 124x + a It is given that the function f... View Article
The set of extreme values of function f (x) = tan x, x ∈ R – {(2k + 1) (π / 2)) , k ∈ I} 1) ɸ 2) R - (-1, 1) 3) R+ ⋃ {0} 4) R Solution: (1) ɸ f (x) = tan x f’ (x) = sec2 x f’ (x) = 0 sec2 x = 0 It is not possible.... View Article
Let f (x) = x2e-2x, x > 0. Then the maximum value of f (x) is 1) 1 / e 2) 1 / 2e 3) 1 / e2 4) 4 / e4 Solution: (3) 1 / e2 f (x) = x2 e-2x f’ (x) = e-2x (2x) – (2x2) e-2x f’ (x) = 2e-2xx... View Article
The minimum value of the value of the function 2 cos 2x – cos 4x in 0 ≤ x ≤ π is 1) 0 2) 1 3) 3 / 2 4) –3 Solution: (4) –3 y = 2 cos 2x – cos 4x = 2 cos 2x (1 - cos 2x) + 1 = 4 cos 2x sin2 x + 1 sin2 x ≥ 0... View Article
If sum of two numbers is 3, then maximum value of the product of first and the square of second is 1) 4 2) 3 3) 2 4) 1 Solution: (1) 4 Assume the first number to be 3 − x and the second number be x. f (x) = (3 - x)x2 = 3x2 - x3 ⇒ f'... View Article
A minimum value of integral from 0 to x te-t2 dt is 1) 1 2) 2 3) 3 4) 0 Solution: (4) 0 f (x) = ∫0x te-t^2 dt f’ (x) = xe-x^2 = 0 x = 0, f’’ (x) = e-x^2 (1 - 2x2) f’’ (0) = 1 >... View Article
The function f (x) = |px – q| + r |x|, x ∈ (- ∞, ∞) where p > 0, q > 0, r > 0 assumes its minimum value only at one point if 1) p ≠ q 2) q ≠ r 3) r ≠ p 4) p = q = r Solution: (3) r ≠ p f (x) = | px − q| + r |x|, x ∈ (− ∞, ∞) Thus, f has infinite... View Article
The largest term in the sequence an = n2 / [n3 + 200] is given by 1) 529 / 49 2) 8 / 89 3) 49 / 543 4) None of these Solution: (3) 49 / 543 an = n2 / [n3 + 200] = f (x) f’ (x) = x [(400 - x3) /... View Article
The value of a so that the sum of the squares of the roots of the equation x2 – (a – 2) x – a – 1 = 0 assume the least value, is 1) 2 2) 1 3) 3 4) 0 Solution: (2) 1 Let α and β be the roots of equation x2 - (a - 2)x - a - 1 = 0 α + β = a - 2 and αβ = - a - 1... View Article
xx has a stationary point at 1) x = e 2) x = 1 / e 3) x = 1 4) x = √e Solution: (2) x = 1 / e y = xx ⇒ logy = x (logx), (x > 0) dy / dx = xx (1 + logx)... View Article
The least value of the sum of any positive real number and its reciprocal is 1) 1 2) 2 3) 3 4) 4 Solution: (2) 2 f (x) = x + (1 / x) ⇒ f' (x) = 1 - (1 / x2) ⇒ f' (x) = 0 ⇒ x2 - 1 = 0 ⇒ x = 1, - 1 x is... View Article
If for a function f(x), f’(a) = 0, f’’(a) = 0, f’’’(a) > 0 then x = a, f(x) is 1) Minimum 2) Maximum 3) Not an extreme point 4) Extreme point Solution: (3) Not an extreme point f’ (a) = 0 f’’ (a) = 0 a is... View Article
The number that exceeds its square by the greatest amount is 1) –1 2) 0 3) 1 / 2 4) 1 Solution: (3) 1 / 2 Assume the number to be x. Then, y = x - x2 dy / dx = 1 - 2x and d2y / dx2 = - 2... View Article
The greatest value of f (x) = (x + 1)⅓ – (x – 1)⅓ on [0, 1] is 1) 1 2) 2 3) 3 4) 4 Solution: (2) 2 f (x) = (x + 1)⅓ - (x - 1)⅓ f’ (x) = (1 / 3) [1 / (x + 1)⅔ - 1 / (x - 1)⅔] = [(x + 1)⅔ -... View Article
The sum of two numbers is fixed. Then its multiplication is maximum, when 1) Each number is half of the sum 2) Each number is 1 / 3 and 2 / 3 respectively of the sum 3) Each number is 1 / 4 and 3 / 4 respectively... View Article
Maximum value of (1 / x)x is 1) (e)e 2) (e)1/e 3) (e)-e 4) (1 / e)e Solution: (2) (e)1/e f (x) = (1 / x)x f’ (x) = (1 / x)x (log (1 / x) - 1)) f’ (x) = 0... View Article
