which are the following are true?

Question

if F(x)=∫₀^x (t-1)/(t²+9)dt, then which of the following statements are true?

I. function F has a minimum value at x=1

II. function F has increase on the (1,∞)

III. F’’(x) is negative on (-1,3)

A)I and II, only B)I and III, only C)II, only D) I, only

the answer is A i just dont know how to get it

Answer 1

F(x) = ∫_{(from 0 to x)}[(t-1)/(t²+9)]dt

 

By the Fundamental Theorem of Calculus, F'(x) = (x-1)/(x²+9)

 

    F'(x) = 0 when x-1 = 0 (i.e., x = 1)

 

    When x < 1, F'(x) < 0.  So, F(x) is decreasing 

    When x > 1, F'(x) > 0.  So, F(x) is increasing

 

    Therefore, l and ll are both true

 

    F"(x) = [(x²+9)(1) - 2x(x-1)] / (x²+9)²

 

            = (-x² + 2x + 9) / (x² + 9)²

 

    F"(0) = 1/9   So, lll is FALSE.

 

ANSWER:  A