Question

suppose that ( f ) is a continuous function and the following information is known:
( f(2)=2, f(4)=-1, f(6)=6, f(8)=4, f(10)=-3 ).
let
( f(x)=int_{3 x}^{x^{3}} f(t) d t ).
calculate ( f^{prime}(2) ).
-8
30
-2
10
1

Explanation:

Step1: Apply Leibniz rule to F(x)

$F'(x) = f(x^3) \cdot 3x^2 - f(3x) \cdot 3$

Step2: Substitute x=2 into F'(x)

$F'(2) = f(2^3) \cdot 3(2)^2 - f(3\cdot2) \cdot 3 = f(8) \cdot 12 - f(6) \cdot 3$

Step3: Use given f values to calculate

$f(8)=4, f(6)=6$, so $F'(2)=4\cdot12 - 6\cdot3 = 48 - 18 = 30$

Answer:

30