Let f : R → 0 , 1 , be a continuous function then, which of the following function(s), has(have) the values zero at some point in the interval, (0,1)?

Step-by-step solution

For option (i), Let M x = x - ∫ 0 π 2 - x f t · cos t   d t ∴   M 0 = 0 - ∫ 0 π 2 f t . cos t   d t < 0 Also, M 1 = 1 - ∫ 0 π 2 - 1 f t . cost  d t > 0 ⇒       M 0 .   M 1 < 0 (so M ( x ) can be zero) For option (ii), Let, k x = x 9 - f x Now, k 0 = - f 0 < 0 , as   f ∈ 0 ,   1 Also, k 1 = 1 - f 1 > 0       A s   f ∈ 0,1 ⇒          k 0