12th std. Rolle's theorem

[00:00:00][music] [music] [music] >> Hello sub log. Welcome back to Devsar's Mathematics.

[00:00:20]Today is the day for 12th standard.

[00:00:23]12th standard may kya dekhne wale ho?

[00:00:25]So, let's see.

[00:00:27]We are going to have a look at Rolle's theorem.

[00:00:32]Which is present in applications of derivatives.

[00:00:35]Okay, applications of derivatives may Rolle's theorem karke hai.

[00:00:39]So, there is an important problem based on that.

[00:00:43]Now, Rolle was a mathematician in the 16th century who got this derived this Rolle's theorem.

[00:00:50]Okay? It's in the name of Mr. Rolle who was a very ancient mathematician.

[00:00:56]What contribution he has maths ke liye?

[00:00:59]So, today we will see his theorem and see uh how to get certain values which are asked. Okay? So, all set?

[00:01:09]Right. So, this is again applications of derivatives may hai.

[00:01:14]To dekhenge kya hai.

[00:01:17]Exercise 2.3 question six.

[00:01:20]M2.

[00:01:22]The function f(x) x into x + 3 into e raise e raise to x upon 2 satisfies all condition of Rolle's theorem on -3,0. This is called as a closed interval.

[00:01:39]Okay? Find the value of c such that f'(c) is equal to zero.

[00:01:45]Okay, I'll explain everything. Batata hu mai.

[00:01:48]To key step may likha hai, get the equation c squared minus c minus six equal to zero.

[00:01:56]C ki value puchi hai.

[00:01:59]The at me. There is one important thing that when Rolle's theorem is satisfied there is always a number C such that f dash C is equal to zero.

[00:02:15]Even if it is not given f of B f dash C equal to zero follows.

[00:02:27]And that value of C should lie in this given interval.

[00:02:33]Closed interval coming here.

[00:02:36]Minus three select zero to all numbers in between. Minus three, minus two, minus one, one, zero including minus three and zero. Okay, all all numbers in between real numbers.

[00:02:51]Right? So, you have to find a number C such that Clear?

[00:03:02]Right.

[00:03:07]If starting to look at what I'm I'm going to solve, don't worry. Okay, I'm not I'm not just going to read this out to you. Starting to easy steps I have written down so that time is saved.

[00:03:22]The given function satisfies Rolle's theorem means there exists a number C such that f dash C equal to zero.

[00:03:35]To find f dash C about to find f dash C, we first find f dash x.

[00:03:41]f dash x equal to f dash x f dash C million so how do we start?

[00:03:46]By finding out f dash x.

[00:03:48]Okay, that is how your thinking should be. f dash x find out karne ke baad f dash c automatically milega equation.

[00:03:56]Clear?

[00:03:58]Now to find f dash x, naturally we have to write f x.

[00:04:02]Right? So, steps are pehla f x likho.

[00:04:07]Uske baad mein f dash x find out karo.

[00:04:10]Uske baad mein just replace x by c, you are going to get f dash c.

[00:04:15]Okay? So, what are the steps? Repeat again after me.

[00:04:19]f x Differentiate with respect to x f dash x.

[00:04:25]f dash x ke baad f dash c.

[00:04:29]Chalo abhi, let's start.

[00:04:32]So, this is the value of f x. Yeah, f x ki value di hui hai.

[00:04:37]Abhi kya karne wala? I'll just take x inside. x se multiply kiya.

[00:04:42]x squared x into x x squared plus three into x three x. x andar liya.

[00:04:48]e raise to x by two.

[00:04:50]I'm trying to make the function simpler.

[00:04:53]So, after I get this f x d by d x of f x, differentiate with respect to x. d by d x of f x is equal to d by d x of r h s.

[00:05:06]Iska derivative.

[00:05:09]Abhi important cheez kya hai? You might have guessed.

[00:05:13]x squared plus three x one function hai.

[00:05:16]x e raise to x by two second function hai.

[00:05:20]You have a multiplication sign here.

[00:05:23]So, when you have to take the product derivative of the product derivative of product of two functions, to kya karna hai? Use product rule. Jo 11 standard mein hum log ne seekha hai.

[00:05:37]Derivative of the product of two functions. Yeh pehla function hai, yeh second function hai.

[00:05:42]Baahar d by d x hai. Means kya karne wale ho? Use product rule. Abhi yaad hai 11 standard ka d/dx of fx, we all know f'x.

[00:05:59]First function, derivative of second.

[00:06:02]x^2 + 3x d/dx of e^x/2 First function, derivative of second plus second function d/dx of x^2 + 3x Right? You might be knowing this.

[00:06:19]First function, derivative of second plus second function, derivative of first. Simple.

[00:06:24]So, that's what I have written. And derivative means d/dx. Okay? That's the symbol.

[00:06:36]Let's solve now.

[00:06:47]Okay. First function d/dx of e^x/2. Now, you should know certain formula. Derivative of e^x is e^x. So, derivative of e^x/2 is e^x/2 into d/dx of x/2.

[00:07:25]We know that when we studied derivatives.

[00:07:29]Okay. Plus d/dx of x^2 + 3x We'll do it orally.

[00:07:50]d / dx of x squared is 2x.

[00:07:54]d / dx of x squared 2x basic formula x raised to n formula plus d / dx of 3x d / dx of x is 3 is a constant.

[00:08:14]derivative of x is 1 11 standard stuff here So, d / dx of 3x is 3 into 1, which is 3.

[00:08:25]Clear?

[00:08:28]Followed?

[00:08:29]Okay.

[00:08:30]Next step.

[00:08:49]derivative of x by 2 here How do you find derivative of x by 2 orally?

[00:08:55]1 by 2 constant take it outside Your constant is going to be 1 by 2 outside into d / dx of x, which is 1 So, derivative of x by 2 is half into 1, which is half.

[00:09:12]plus here is my good change in the weather Right?

[00:09:31]Okay, so that is our f dash x.

[00:09:34]Clear? So, let us write it properly now.

[00:09:37]>> write it properly now.

[00:09:46]It come connect here.

[00:09:48]E raised to X by 2 go common level.

[00:09:52]Take it outside.

[00:09:59]E raised to X by 2 bar again.

[00:10:01]What remains X squared plus 3 X?

[00:10:09]Into half coming in here.

[00:10:11]Upon two.

[00:10:14]Into half coming in here. Two denominator.

[00:10:19]Okay.

[00:10:22]Common layer now.

[00:10:24]So we'll take square bracket.

[00:10:31]Look at your E raised to X by 2 bar nickelate. X squared plus 3 X upon 2 into one nickel zero at my head. Plus 2 X plus 3.

[00:10:49]But what my head?

[00:10:51]Mhm. Clear?

[00:10:54]Okay, but I'm going to do F dash X on again but F dash C. So what is the procedure?

[00:11:00]Write F X find F dash X replace X by C to find F dash C F dash C equation zero that they all zero killing here because F dash C equal to zero is always there.

[00:11:12]For all zero when all zero is satisfied.

[00:11:16]To all we replace X by C.

[00:11:34]Jaja X >> X same thing over here. Same thing over Everywhere I have replaced X by C because of which automatically I'm going to have that C value.

[00:11:58]So by conditions of roll's theorem f dash C equal to zero.

[00:12:11]So let us equate all this to zero.

[00:12:15]Equate it to zero.

[00:12:43]Okay.

[00:12:47]One interesting thing e raised to C by 2 common e raised to C by 2 will get vanished.

[00:13:00]Because The next step make your own a while I am here.

[00:13:06]e C This into this bracket.

[00:13:10]So whenever you are seen in linear equations in 10th standard quadratic equations, whenever you take a number outside, term outside If you take it on the right hand side, what happens?

[00:13:20]I'll take it here itself. Look at that.

[00:13:24]You have to say e raised to C by 2 remove करके basic math here.

[00:13:29]I'll be taking over here.

[00:13:32]Upon e raised to c by 2.

[00:13:35]So you have to say guy have over here here again.

[00:13:38]Zero upon anything zero.

[00:13:41]So e raised to c by 2 has gone. Guy have over here and we don't need that extra thing. Okay? So zero upon e raised to c by 2 is zero. That I will write in the next step and then we will solve.

[00:13:55]Tell them what I have to be called a B.

[00:13:57]It's a common word here.

[00:14:02]Let's solve the equation.

[00:14:12]Right.

[00:14:14]So zero upon anything is zero.

[00:14:17]So what remains?

[00:14:22]c squared plus 3c upon 2 plus 2c plus 3 equal to zero.

[00:14:38]e raised to c by 2 guy have over here.

[00:14:40]I'll be I cannot directly cross multiply by 2.

[00:14:45]You have to find the LCM.

[00:14:47]Okay? Find the LCM or you can multiply each term by 2.

[00:14:53]Basic math. I want to get rid of this 2.

[00:14:56]Let us multiply each term by 2.

[00:14:59]So it's going to 2 say multiply by 2.

[00:15:11]It's going to 2 say multiply by 2.

[00:15:17]It's going to 2 say multiply by 2.

[00:15:19]Equation means sub that say sub that say 2 say multiply by 2.

[00:17:16]>> 10 standard stuff 7 C 4 6 C + 1 C Take something common when you go to this is C common in C + 6 If nothing is common, one is always common.

[00:17:32]This bracket gets repeated.

[00:17:37]Then C + 6 again goes outside.

[00:17:52]Get the answer now.

[00:17:57]C + 6 equal to 0 gives you C equal to minus 6.

[00:18:04]C + 1 equal to 0 gives you C equal to minus 1.

[00:18:10]Mhm?

[00:18:10]Clear?

[00:18:13]So, I'll be a interval check here. Last step.

[00:18:20]Minus 3 here there.

[00:18:24]Minus 2 minus 1 0 1 Up to 0 here.

[00:18:36]So, if you check both the values now, minus 6 will lie somewhere over here.

[00:18:42]Minus 4 minus 5 minus 6 So, minus 6 does not belong to this interval. It's going to be like that minus 6. So, minus 6 will be like that minus 6. You have to select the value which fits in that interval. C equal to minus 1 will be like that minus 1 better here.

[00:19:01]So, this is our correct answer.

[00:19:03]C equal to minus 1 or how do you write that?

[00:19:10]C belongs to where is the symbol for belongs to.

[00:19:16]Minus 3 0.

[00:19:23]Followed? This is how you use Rolle's theorem in finding the value of C.

[00:19:31]FX F dash X F dash C put F dash C equal to zero get a quadratic equation solve karo.

[00:19:37]This was previously asked in board exam or CET maybe which get the value of C.

[00:19:42]Four options were given. Okay. What was that enjoy share it with your friends.

[00:19:48]Okay.

[00:19:49]And let me know if you have got any doubts. That will be all for today. Bye.

[00:19:54]See you.

[00:19:57]>> [music] [music] [music]