Let g x = ∫ 0 x f t d t , where f is continuous function in [ 0 , 3 ] such that 1 3 ≤ f t ≤ 1 for all t ∈ [ 0 , 1 ] and 0 ≤ f t ≤ 1 2 for all t ∈ ( 1 , 3 ] . The largest possible interval in which g ( 3 ) lies is :

Question

Let g x = &#8747; 0 x f t d t , where f is continuous function in [ 0 , 3 ] such that 1 3 &#8804; f t &#8804; 1 for all t &#8712; [ 0 , 1 ] and 0 &#8804; f t &#8804; 1 2 for all t &#8712; ( 1 , 3 ] . The largest possible interval in which g ( 3 ) lies is :

Aicharya · Accepted Answer

1 3 &#8804; f t &#8804; 1 &#8704; t &#8712; 0 , 1 0 &#8804; f t &#8804; 1 2 &#8704; t &#8712; ( 1 , 3 ] Now, g 3 = &#8747; 0 3 f t d t = &#8747; 0 1 f t d t + &#8747; 1 3 f t d t &#8757; &#8747; 0 1 1 3 d t &#8804; &#8747; 0 1 f t d t &#8804; &#8747; 0 1 1 . d t &#160;&#160; . . . 1 and &#8747; 1 3 0 d t &#8804; &#8747; 1 3 f 1 d t &#8804; &#8747; 1 3 1 2 d t &#160;&#160; . . . 2 Adding, we get 1 3 + 0 &#8804; g 3 &#8804; 1 + 1 2 3 - 1 1 3 &#8804; g 3 &#8804; 2

Let g x = ∫ 0 x f t d t , where f is continuous function in [ 0 , 3 ] such that 1 3 ≤ f t &#8804…

Solution

Let g x = &#8747; 0 x f t d t , where f is continuous function in [ 0 , 3 ] such that 1 3 &#8804; f t &#8804…

Solution

Let g x = ∫ 0 x f t d t , where f is continuous function in [ 0 , 3 ] such that 1 3 ≤ f t &#8804…