If F and f are differentiable functions such that F ( x )=∫ limits0 x f ( t ) dt, and if F ( a )=-2 and F ( b )=-2 where a

Question

If F and f are differentiable functions such that F ( x )=∫ limits0 x f ( t ) dt, and if F ( a )=-2 and F ( b )=-2 where a < b, then which of the following must be true? (A) f ( x )=0 for some x such that a < x < b. (B) f ( x )<0 for all x such that a < x < b. (C) F ( x ) ≤ 0 for all x such that a < x < b. (D) F ( x )=0 for some x such that a < x < b.

Tardigrade Admin · Accepted Answer

Use Rolle's Theorem for F ( x ) in [ a , b ] ; F ( a )= F ( b ), also f is differentiable Hence, we have F prime(x)=f(x)=0 ⇒ continuous [12th, 26-08-2012, P-1] for some x ∈(a, b) ⇒ A

Q. If $F$ and $f$ are differentiable functions such that $F (x) = 0 \int x f (t) d t$ , and if $F (a) = - 2$ and $F (b) = - 2$ where $a < b$ , then which of the following must be true?

$f (x) = 0$ for some $x$ such that $a < x < b$ .

$f (x) < 0$ for all $x$ such that $a < x < b$ .

$F (x) \leq 0$ for all $x$ such that $a < x < b$ .

$F (x) = 0$ for some $x$ such that $a < x < b$ .

Use Rolle's Theorem for $F (x)$ in $[a, b]; F (a) = F (b)$ , also $f$ is differentiable
Hence, we have $F^{'} (x) = f (x) = 0 \Rightarrow$ continuous
[12th, 26-08-2012, P-1] for some $x \in (a, b) \Rightarrow A$

Q. If F and f are differentiable functions such that F(x)=0∫x​f(t)dt, and if F(a)=−2 and F(b)=−2 where a<b, then which of the following must be true?

f(x)=0 for some x such that a<x<b.

f(x)<0 for all x such that a<x<b.

F(x)≤0 for all x such that a<x<b.

F(x)=0 for some x such that a<x<b.