KVPY Government Jobs Exams

Let f : R → R be a differentiable function such that f (a) = 0 = f (b) and f ′(a) f ′(b) > 0 for some a < b. Then
the minimum number of roots of f ′(x = 0 in the interval (a, b) is-
(A) 3 (B) 2 (C) 1 (D) 0

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KVPY Other Question

The product (1 + tan 1º) (1 + tan 2º) (1 + tan 3º) …. (1 + tan 45º) equals- The roots of (x – 41)^49 + (x – 49)^41 + (x – 2009)^2009 = 0 are -
(A) all necessarily real
(B) non-real except one positive real root
(C) non-real except three positive real roots
(D) non-real except for three real roots of which exactly one is positive