Let , where is continuous on the closed interval . Answer the…

MATHEMATICS · Multi-part · CBSE · Mathematics · Integrals · Integrals

Problem

Let

F(x)=\displaystyle \int_{a}^{x} f(t)\,dt

, where

f

is continuous on the closed interval

[a,b]

. Answer the following: (i) State the First Fundamental Theorem of Calculus. (ii) Explain, with reasons, how the sign of

f(x)

on an interval

(p,q)\subset(a,b)

determines whether

F(x)

is increasing or decreasing on that interval. (iii) Suppose

f(x)\ge 0

for

1<x<4

and

f(x)\le 0

for

4<x<6

, with

f

continuous on

[1,6]

. Describe the behaviour of

F(x)=\displaystyle \int_{1}^{x} f(t)\,dt

(1,4)

and on

(4,6)

, and comment on the nature of the point

x=4

for the function

F

. Justify your answer.