Let f x be a derivable function in -0-3- such that f 0-0- f 3-3 and f - x - 1 - x -0-3- Then the value of f 1 is 1A- n-2B- 3-3C- 1D- Cannot be determined

The correct option is C 1LMVT on f(x) in [0,1] f(1) f(0)1 0=f'(c_1)≤1 ⇒ f(1)≤1......(1) LMVT on f(x) in [1,3] f(3) f(1)3 1=f'(c_2)≤1 ⇒ f(1)≥1......(2) From ( ...

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Byju's Answer

Standard XII

Mathematics

Global Maxima

Let f x be a ...

Question

Let $f (x)$ be a derivable function in $[0, 3]$ such that $f (0) = 0, f (3) = 3$ and $f^{'} (x) \leq 1 \forall x \in (0, 3)$ . Then the value of $f (1)$ is

\frac{1}{2}

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\frac{1}{3}

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Cannot be determined

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Solution

The correct option is B $1$
LMVT on $f (x)$ in $[0, 1]$
$\frac{f (1) - f (0)}{1 - 0} = f^{'} (c_{1}) \leq 1 \Rightarrow f (1) \leq 1...... (1)$

LMVT on $f (x)$ in $[1, 3]$
$\frac{f (3) - f (1)}{3 - 1} = f^{'} (c_{2}) \leq 1 \Rightarrow f (1) \geq 1...... (2)$

From $(1)$ and $(2)$
$f (1) = 1$

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Similar questions

Q. Let

f (x)

b ea function defined on

R

such that

f^{'} (x) = f^{'} (3 - x), \forall x \in [0, 3]

with

f (0) = - 32

and

f (3) = 46

, then find the value of

\int_{0}^{3} f (x) d x

Q. Let

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be a twice differentiable function defined on

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such that

f (0) = 1,

f^{'} (0) = 2

and

f^{'} (x) \neq 0

for all

x \in R .

∣ ∣ ∣ \begin{matrix} f (x) & f^{'} (x) f^{'} (x) & f^{''} (x) \end{matrix} ∣ ∣ ∣ = 0,

for all

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then the value of

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Q. Consider a function

f (x)

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f^{'} (x)

f^{''} (x)

exists for all

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and that

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, then which of the following is/are correct.

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Let f(x) be a derivable function in [0,3] such that f(0)=0,f(3)=3 and f′(x)≤1 ∀x∈(0,3). Then the value of f(1) is

The correct option is B 1LMVT on f(x) in [0,1] f(1)−f(0)1−0=f′(c1)≤1⇒f(1)≤1......(1) LMVT on f(x) in [1,3] f(3)−f(1)3−1=f′(c2)≤1⇒f(1)≥1......(2) From (1) and (2) f(1)=1

Let $f (x)$ be a derivable function in $[0, 3]$ such that $f (0) = 0, f (3) = 3$ and $f^{'} (x) \leq 1 \forall x \in (0, 3)$ . Then the value of $f (1)$ is

The correct option is B $1$
LMVT on $f (x)$ in $[0, 1]$
$\frac{f (1) - f (0)}{1 - 0} = f^{'} (c_{1}) \leq 1 \Rightarrow f (1) \leq 1...... (1)$

LMVT on $f (x)$ in $[1, 3]$
$\frac{f (3) - f (1)}{3 - 1} = f^{'} (c_{2}) \leq 1 \Rightarrow f (1) \geq 1...... (2)$

From $(1)$ and $(2)$
$f (1) = 1$