Maths Applications of Derivatives

Let z₁, z₂ be the roots of the equation z² + az + 12 = 0 and z₁, z₂ form an equilateral triangle with origin. Then, The value of |a| is ______.

If the point P on the curve, 4x² + 5y² = 20 is farthest from the point Q(0, −4) then PQ² is equal to:

If the minimum and the maximum values of the function f: [-π/2, π/2] → R, defined by f(θ) = |-sin²θ -1 -sin²θ; -cos²θ -1 -cos²θ; 12 10 -2| are m and M respectively, then the ordered pair (m, M) is equal to:

The area (in sq. units) of the largest rectangle ABCD whose vertices A and B lie on the x-axis and vertices C and D lie on the parabola, y = x² – 1 below the x-axis, is:

Let f be a twice differentiable function on (1,6). If f(2) = 8, f'(2) = 5, f'(x) ≥ 1 and f''(x) ≥ 4, for all x ∈ (1,6), then:

If Lim(x→0) [(tan x)/x]^(1/x²) = p, then 96log? p is equal to

If the tangent to the curve, y = eˣ at a point (c, eᶜ) and the normal to the parabola, y² = 4x at the point (1,2) intersect at the same point on the x-axis, then the value of c is

If the surface area of a cube is increasing at a rate of 3.6 cm²/sec, retaining its shape; then the rate of change of its volume (in cm³/sec ), when the length of a side of the cube is 10 cm, is:

The function, f(x) = (3x – 7)x²/³, x ∈ R, is increasing for all x lying in: