Let the function ,f:-7,0→R be continuous on -7,0 and differentiable on -7,0. If f-7=-3 and f^(')x≤2 for all x∈-7,0, then for all such functions f, f-1+f0 lies in the interval

Correct answer: (A) -∞,20. Lets use LMVT for x∈-7,-1 (f-1-f-7)/(-1+7)≤2 (f-1+3)/6≤2⇒f-1≤9 Lets use LMVT for x∈-7,0 (f0-f-7)/(0+7)≤2 (f0+3)/7≤2⇒f0≤11 ∴f0+f-1≤20

Q46 / 75 · JEE Main 2020 (07 Jan Shift 1) · Mathematics

Let the function $, f : [- 7, 0] \to R$ be continuous on $[- 7, 0]$ and differentiable on $(- 7, 0) .$ If $f (- 7) = - 3$ and $f^{'} (x) \leq 2$ for all $x \in (- 7, 0),$ then for all such functions $f, f (- 1) + f (0)$ lies in the interval

Correct answer
A
$(- \infty, 20]$
B
$[- 3, 11]$
C
$(- \infty, 11]$
D
$[- 6, 20]$

Solution

Lets use LMVT for

x \in [- 7, - 1]

\frac{f (- 1) - f (- 7)}{(- 1 + 7)} \leq 2

\frac{f (- 1) + 3}{6} \leq 2 \Rightarrow f (- 1) \leq 9

Lets use LMVT for

x \in [- 7, 0]

\frac{f (0) - f (- 7)}{(0 + 7)} \leq 2

\frac{f (0) + 3}{7} \leq 2 \Rightarrow f (0) \leq 11

∴ f (0) + f (- 1) \leq 20

Hence, the correct answer is option: 'A'.

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Let the function ,f:-7,0→R be continuous on -7,0 and differentiable on -7,0. If f-7=-3 and f'x≤2 for all x∈-7,0, then for all such functions f, f-1+f0 lies in the interval

Solution

Relevant Questions

Let the function $, f : [- 7, 0] \to R$ be continuous on $[- 7, 0]$ and differentiable on $(- 7, 0) .$ If $f (- 7) = - 3$ and $f^{'} (x) \leq 2$ for all $x \in (- 7, 0),$ then for all such functions $f, f (- 1) + f (0)$ lies in the interval