GATE CSE 2021 Set 2 | Question: 25

Question

Suppose that $f: \mathbb{R} \rightarrow \mathbb{R}$ is a continuous function on the interval $[-3, 3]$ and a differentiable function in the interval $(-3,3)$ such that for every $x$ in the interval, $f’(x) \leq 2$. If $f(-3)=7$, then $f(3)$ is at most __________

Answer 1

Given that $f’(X) \leq 2$ and $f(-3) = 7$,

As maximum slope is positive and we need value of $f(x)$ at $3$ which is on right side of $-3$,  we can assume $f(x)$ as a straight line with slope $2$. It will give us the correct result.

Let $f(x) = 2x + b,$

$f(-3) = -6 + b = 7 \Rightarrow b = 13$

$f(x) = 2x + 13$

$f(3)_{\max} = 6 + 13 = 19.$

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Correct method would be using Mean Value Theorem. Above method will work only if you can analyze the cases correctly and can assume the $f(x)$ without any loss of accuracy, otherwise you are very prone to commit a mistake that way.

Using Mean Value Theorem:

$f’(c) = \frac{f(b) – f(a)}{b -a }$

     $\Rightarrow f’(c) = \frac{f(3) – f(-3)}{3 -(-3)} =  \frac{f(3) – 7}{6} $

     $\Rightarrow f(3) = 6f’(c) +7 $

As $f’(c)\leq 2$,

     $\Rightarrow f(3) \leq 6*2 +7 = 19.$

Comments

Single Step Solution:-

Given, $f’(x)<=2$

[Integrating Both Sides]

$\implies \int f'(x)dx <= \int 2\ dx$

$ \implies f(x)<=2x+c $          -------->$ \ (1)$

Given, $f(-3)=7$, so putting in eqn. $(1)$

$f(-3)<=2(-3)+c $

$\implies 7<=-6+c $

$\implies c>=13$

$\therefore f(3)<= 2(3)+13 $

$\implies f(3)<=19$ ($Ans$)

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@Abhrajyoti00how did u get the idea of integrating both sides ?

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@shikhar500 I don’t know why, but I always love to integrate derivatives :) Jokes apart. In this q, f’(x) is asked and we need to find f(x). So definitely integration will help.

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@Abhrajyoti00 Bro……….. Tussi Great ho……..🙂 ekdum mast solution hai. 

Answer 2

The answer is $19$. Check the image below.

 

Comments

from where you study calculus for gate please tell resources

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Gilbert Strang Calculus book

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Thanks

Answer 3

So we are given the max angle as 2 .

We know tan(theta) = height/base

 

So, 2= height / (3 - (-3))

Therefore : height = 2*6 = 12

And we know the level is at .. 7 so answe is 12+7 = 19