F (x) Is A Polynomial Of Degree 4 With Real Coefficients Such That f (x) = 0 Is Satisfied By x = 1, 2, 3 Only, Then f’(1). f’(2). f’(3) Is Equal To:

Let f (x) and g (x) be differentiable for 0 ≤ x ≤ 2 such that f (0) = 2, g(0) = 1 and f (2) = 8. Let there exists a real number c in [0, 2] such that f’(c) = 3g’(c) then the value of g(2) must be:

If f (x) = log_e x and g(x) = x² and c Ïµ (4, 5), then  is equal to: 

In [0, 1] lagrange’s mean value theorem is not applicable to

Let f (x) satisfy the requirement of lagrange’s mean value theorem in [0, 2]. If f (0) and   

Let f : [2, 7] and [0, ∞) be a continuous and differentiable function.

Then the value of (f (7) – f (2))  is (where c Ïµ (2, 7))

The equation sin x + x cos x = 0 has at least one root in the interval

Let f (x) = ax⁵ + bx⁴ + cx³ + dx² + ex, where a, b, c, d, e Ïµ R and f (x) = 0 has a positive root α, then 

Between any two real roots of the equation e^x sin x – 1 = 0, the equation e^xcos x + 1 = 0 has

If f (x) is a polynomial of degree 5 with real coefficients such that  has 8 real roots then f (x) = 0 has: