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The minimum value of f (x) = |3 – x| + 7 is

1) 0 2) 6 3) 7 4) 8 5) 10 Solution: (3) 7 f (x) = |3 – x| + 7 f (x) = {(3 - x + 7), (3 - x) ≥ 0 and - (3 - x) + 7, when 3 - x... View Article

The value of a for which the function f (x) = a sin x + (1 / 3) sin 3x has an extremum at x = π / 3 is

1) 1 2) –1 3) 0 4) 2 Solution: (4) 2 f (x) = a sin x + (1 / 3) sin 3x f’ (x) = 0 f’ (x) = a cos x + cos 3x f’ (π / 3) = 0 (a /... View Article

Let f’’R → R be defined by f (x) = {k – 2x, if x ≤ – 1 and 2x + 3, if x > – 1. If f has a local minimum at x = – 1, then a possible value of k is

1) 1 2) 0 3) - 1 / 2 4) –1 Solution: (4) –1 f (x) = {k - 2x, if x ≤ - 1 and 2x + 3, if x > - 1 limx→1+ f (x) = 1 As f (-1) =... View Article

f (x) = x⁵ – 5x⁴ + 5x³ + 1 has

1) Two maximum and two minimum value 2) Two maximum and one minimum value 3) One maximum and one minimum value 4) None of the above... View Article

The minimum value of (x² + (250 / x)), is

1) 75 2) 50 3) 25 4) 55 Solution: (1) 75 y = f (x) = (x2 + (250 / x)) dy / dx = f'(x) = 2x - [250 / x2] Put f (x) = 0 2x3 - 250... View Article

The function y = a (1 – cos x) is maximum when x =

1) π 2) π / 2 3) - π / 2 4) - π / 6 Solution: (1) π y = a (1 - cos x) y' = a sin x y' = 0 sinx = 0 x = 0, π y'' = a cos x... View Article

If PQ and PR are the two sides of a triangle, then the angle between them which gives maximum area of the triangle is

1) π 2) π / 3 3) π / 4 4) π / 2 Solution: (4) π / 2 Assume PQ = a and PR = b. Δ = (1 / 2) ab sinθ (- 1 ≤ sinθ ≤ 1) The area is... View Article

If ab = 2a + 3b, a > 0, b > 0 then the minimum value of ab is

1) 12 2) 24 3) 1 / 4 4) None of these Solution: (2) 24 ab = 2a + 3b (a - 3) b = 2a b = 2a / [a - 3] z = ab = [2a2] / [a - 3] dz / da = 2 [(a... View Article

The maximum value of xy = 8 is

1) 20 2) 16 3) 24 4) 8 5) 4 Solution: (4) 8 x + 2y = 8 y = [8 - x] / 2 f (x) = xy = x ([8 - x] / 2) = 4x - (x2 / 2) For... View Article

On [1, e] the greatest value of x² log x

1) e2 2) (1 / e) log (1 / √e) 3) e2 log √e 4) None of these Solution: (1) e2 f (x) = x2 log x f' (x) = (2 logx + 1)x Put f' (x)... View Article

In (- 4, 4), the function f (x) = Integral from -10 to x (t⁴ – 4) e^-4t dt has

1) No extrema 2) One extremum 3) Two extrema 4) Four extrema Solution: (3) Two extrema f (x) = ∫-10x (t4 - 4) e-4t dt f’ (x) =... View Article

If y = a log x + bx² + x has its extremum value at x = – 1 and x = 2, then (a, b) =

1) (1, 1 / 2) 2) (1 / 2, 2) 3) (2, - 1 / 2) 4) (- 2 / 3, - 1 / 6) Solution: (3) (2, - 1 / 2) y = a log x + bx2 + x has its... View Article

The perimeter of a sector is p. The area of the sector is maximum when its radius is

1) √p 2) 1 / √p 3) p / 2 4) p / 4 Solution: (4) p / 4 A = (1 / 2) r2θ (1) Length of the arc is s = rθ θ = s / r The perimeter... View Article

If f (x) = x + (1 / x), x > 0 then its greatest value is

1) –2 2) 0 3) 3 4) None of these Solution: (4) None of these f (x) = x + [1 / x] f' (x) = 1 - [1 / x2] Put f (x) = 0, x = −1, 1.... View Article

If f (x) = 1 / [4x² + 2x + 1], then its maximum value is

1) 4 / 3 2) 2 / 3 3) 1 4) 3 / 4 Solution: (1) 4 / 3 f (x) = 1 / [4x2 + 2x + 1] f’ (x) = - (8x + 2) / (4x2 + 2x + 1)2 Put f’... View Article

The function f (x) = 2x³ – 3x² – 12x + 4 has

1) No maxima and minima 2) One maximum and one minimum 3) Two maxima 4) Two minima Solution: (2) One maximum and one minimum f (x) = 2x3 -... View Article

Differential coefficient of √(sec√x) is

(1) (1/4√x)(sec√x)3/2sin √x (2) (1/2√x)(sec√x)3/2sin √x (3) (1/2√x)(sec√x)sin √x (4) none of these Solution: Let y = √(sec√x)... View Article

For the curve √x + √y = 1, dy/dx at (¼, ¼) is

(1) 1 (2) ½ (3) -1 (4) 2 Solution: Given √x + √y = 1 Differentiate w.r.t.x 1/2√x + (1/2√y)dy/dx = 0 dy/dx = (-1/2√x )/(1/2√y)... View Article

If y = (cos x²)², then dy/dx is equal to

(1) -4x sin 2x2 (2) -x sin x2 (3) -2x sin 2x2 (4) -x cos 2x2 Solution: Given y = (cos x2)2 Differentiate w.r.t.x dy/dx = 2 cos... View Article

If y = (x+1)(x+2)(x+3)(x+4)(x+5), then the value of dy/dx at x = 0 is equal to

(1) 742 (2) 472 (3) 374 (4) 274 Solution: Given y = (x+1)(x+2)(x+3)(x+4)(x+5) Differentiate w.r.t.x dy/dx = (x+2)(x+3)(x+4)(x+5) +... View Article

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