If (x) = x _ 3 ^x ( k t - 4 '(t) ) dt , x ( 0, 2 ) and ' ( 3 ) = 3 , then the value of k is equal to

Step-by-step solution

Given equation is: (x) = x _ /3 ^x ( k t - 4 '(t) ) dt Differentiating both sides with respect to x using the product rule and Leibniz rule, we get: '(x) = x x _ /3 ^x ( k t - 4 '(t) ) dt + x ( k x - 4 '(x) ) Substitute x = 3 into the equation. The integral term becomes zero because the upper and lower limits are the same: ' ( 3 ) = 0 + ( 3 ) ( k ( 3 ) - 4 ' ( 3 ) ) We are given ' ( 3 ) = 3 . Also, ( 3 ) = 2 and ( 3 ) = 3 2 . Substituting these values: 3 = 2 ( k ( 3 2 ) - 4 3 ) 3 = k 3 - 8 3 Dividing the entire equ