Let and be two differentiable functions on (the set of all real numbers satisfying and )

MATHEMATICS · Single correct · AP · Calculus AB · Integration and Accumulation of Change · Integration and Accumulation of Change

Problem

Let

f(x)

and

g(x)

be two differentiable functions on

R

(the set of all real numbers satisfying

f(x)=\frac{x^3}{2}+1-x\int_0^x g(t)\,dt

and

g(x)=x-\int_0^1 f(t)\,dt

). Also

L_1

and

L_2

are the tangent lines drawn to the curve

y=f(x)

and passes through the point

(a,g(a))

on the curve

y=g(x)

. Value of

\int_0^1 f(t)\,dt

lies in the interval (a)

(0,\frac{1}{2})

(b)

(\frac{1}{2},1)

(c)

(1,\frac{4}{3})

(d)

(\frac{4}{3},\frac{5}{3})