Let f ( x ) satisfy the requirements of Lagrange's mean value theorem in [0,2]. If f (0)=0 and f '( x ) ≤ (1/2) for all x in [0, 2], then

Question

Let f ( x ) satisfy the requirements of Lagrange's mean value theorem in [0,2]. If f (0)=0 and f '( x ) ≤ (1/2) for all x in [0, 2], then (A) | f ( x )| ≤ 2 (B) f ( x ) ≤ 1 (C) f(x)=2 x (D) f(x)=3 for atleast one x in [0,2]

Tardigrade Admin · Accepted Answer

Applying Lagrange's mean value theorem to

f (x)

in

[0, x], 0 < x \leq 2

, we get

\frac{f ( x ) - f ( 0 )}{x - 0} = f^{'} (c)

for some

c \in (0, 2)

\Rightarrow \frac{f ( x )}{x} \equiv f^{'} (c) \leq \frac{1}{2}

\Rightarrow f (x) \leq \frac{1}{2} x \leq \frac{1}{2} .2 (∵ x \leq 2)

\Rightarrow f (x) \leq 1

Q. Let $f (x)$ satisfy the requirements of Lagrange's mean value theorem in $[0, 2]$ . If $f (0) = 0$ and $f^{'} (x) \leq \frac{1}{2}$ for all $x$ in $[0, 2]$ , then

$∣ f (x) ∣ \leq 2$

$f (x) \leq 1$

$f (x) = 2 x$

$f (x) = 3$ for atleast one $x$ in $[0, 2]$

Q. Let f(x) satisfy the requirements of Lagrange's mean value theorem in [0,2]. If f(0)=0 and f′(x)≤21​ for all x in [0,2], then

∣f(x)∣≤2

f(x)≤1

f(x)=2x

f(x)=3 for atleast one x in [0,2]

Applying Lagrange's mean value theorem to f(x) in [0,x],0<x≤2, we get x−0f(x)−f(0)​=f′(c) for some c∈(0,2) ⇒xf(x)​≡f′(c)≤21​ ⇒f(x)≤21​x≤21​.2(∵x≤2) ⇒f(x)≤1

Q. Let $f (x)$ satisfy the requirements of Lagrange's mean value theorem in $[0, 2]$ . If $f (0) = 0$ and $f^{'} (x) \leq \frac{1}{2}$ for all $x$ in $[0, 2]$ , then

$∣ f (x) ∣ \leq 2$

$f (x) \leq 1$

$f (x) = 2 x$

$f (x) = 3$ for atleast one $x$ in $[0, 2]$