Derivative

Since ,

f ' ( x ) ≤ 1 for all " x " ,

So , f ( x ) ≤ x for all " x " .

Hence , ₀∫¹ f ³ ( x ) dx ≤ ₀∫¹ x ³ dx ≤ 14

Obviously , the numerator is simply " f ( 1 ) " , which cannot be negative since " f ' ( x ) > 0 " and " f ( 0 ) = 0 " .

Hence , the entire expression ≥ 4 f ( 1 )

But maximum value of " f ( 1 ) " is clearly 1 , as " f ( x ) ≤ x " .

Since the expression holds for any " f ( 1 ) " , so it must hold true for " f ( 1 ) = 1 " also .

Hence , it is ≥ 4 .

So , we easily see that the answer is - ( d ) .