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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 a_n = 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

x^x 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

The point (0, 5) is closest to the curve x² = 2y at

1) (2√2, 0) 2) (0, 0) 3) (2, 2) 4) None of these Solution: (4) None of these Let a point on the curve be (h, k). Then h2 = 2k... View Article

The point for the curve y = xe^x

1) x = –1 is minimum 2) x = 0 is minimum 3) x = – 1 is maximum 4) x = 0 is maximum Solution: (1) x = –1 is minimum y = xex For... View Article

If y = {sin (x + a)} / {sin (x + b)}, a ≠ b, then y is

1) Min. at x = 0 2) Max at x = 0 3) Neither min . nor max. at x = 0 4) None of the above Solution: (3) Neither min . nor max. at x =... View Article

Divide 12 into two parts such that the product of the square on one part and the fourth of the second part is maximum

1) 6, 6 2) 5, 7 3) 4, 8 4) 3, 9 Solution: (3) 4, 8 Let the parts be x and 12 – x. f (x) = x4 (12 – x)2, where 0 < x < 12 =... View Article

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