## Vineet Loomba is teaching live on Unacademy Plus

IIT-JEE VIDEO COURSE FOR SURE SHOT SUCCESS IN JEE MAIN AND ADVANCED (IIT-JEE) IIT-JEE MATHS MADE EASY PREPARED BY: ER. VINEET LOOMBA IITian | IIT-JEE MENtOR Search vineet loomba unacademy" on GOOGLE

REVISION COURSE FOR JEE MAIN AND ADVANCED (IIT-JEE) ABOUT ME B.Tech. From IIT Roorkee IIT-JEE Mentor Since 2010 Doubts/Feedback in Comment Section * Comment other topics you want to revise. Follow me @ https://unacademy.com/user/vineetloomba to get updates or search me on Google * Share among your peers as SHARING is CARING!!

APPLICATION OF DERIVATIVES MAXIMA-MINIMA IIT-IEE MATHS MADE EASY PREPARED BY: ER. VINEET LOOMBA IITiAN | IIT-JEE MENTOR FoR SURE SHoT SUcCESS IN JEE MAIN AND ADVANCED (IIT-JEE)

FOR 100% SUCCESS IN JEE MAIN AND ADVANCED (IIT-JEE) Local Maximum highest value in function at this point is greater than its value at all points other than this point in a certain interval (infinitelysmall) defined around this point. In other words, A function f(x) attains a local maximum at x = xo if f(r) >f(x) for all x e (fo,-8, x0 + ), 0 and xx It means f(x) possesses a greatest value at x =x0 in the neighbourhood of f(X) >t when 0 MATHEMATICS FOR IIT-JEE ER. VINEET LOOMBA (IIT RooRKEE)

FOR 100% SUCCESS IN JEE MAIN AND ADVANCED (IIT-JEE) Examples ofLocal Maximum 2 f(x)- sinx f (x) possesses local maximum at x /2 in interval x E [0, 2 ] f(x) 1-x2 f(x)=-(x-1)2/3 f (x) possesses local maximum () possesses local maximum at x = 0 x2 xs0 x-1 >0 local maximum at x-1 1-1x1 x#0 f(x)=e-1x1 f(x) possesses a local maximum at x 0 f(x) f(x) possesses a local maximum at x 0 f(x) possesses a point of MATHEMATICS FOR IIT-JEE ER. VINEET LOOMBA (IIT RooRKee)

FOR 100% SUCCESS IN JEE MAIN AND ADVANCED (IIT-JEE) Local Minimum A function attains a local minimum at a point if value of the function at this point is less than its value at all points other than this point in a certain interval (infinitely small) defined around this point. In other words, A function f (x) attains a local minimum at x =x0 if f(x) <f(x) for all x e (x,-8, x + ), 0 and f(x)x f (x) possesses a local minimum at x = 0 MATHEMATICS FOR IIT-JEE ER. VINEET LOOMBA (IIT RooRKEE)

FOR 100% SUCCESS IN JEE MAIN AND ADVANCED (IIT-JEE) Local Extremum The point at which a function attains either the local maximum value or the local minimum value is known as extreme point or point of local extremum and both local maximum and local minimum values are called the "extreme values" of f(x) or "Local Extremum values". A necessary condition for the existences of a local extremum (either local maximum or local minimum) at point xo ofthe function f() is must be a critical point off(x) i.e. either f)0orf" ) is not defined. MATHEMATICS FOR IIT-JEE ER. VINEET LOOMBA (IIT RooRKEE)

FOR 100% SUCCESS IN JEE MAIN AND ADVANCED (IIT-JEE) FIRST DERIVATIVE TEST Working rule (First derivative test) : Calculate 0 and solve for x and say x - a, b, c etc. Put values of x slightly less than a in and values of x slightly greater than a. If changes sign from positive to negative, then maximum at x -a dx dy dx dy dx If changes sign from nagative to positive, then minimum at x - a. In case there is no change of sign, then neither a maximum nor a minimum. MATHEMATICS FOR IIT-JEE ER. VINEET LOOMBA (IIT RooRKEE)

FOR 100% SUCCESS IN JEE MAIN AND ADVANCED (IIT-JEE) FIRST DERIVATIVE TEST point of local maxima point of non differentiability and point of local maxima f(c1)0 point of non differentiability and point of local minima of local minima C3 C Y' MATHEMATICS FOR IIT-JEE ER. VINEET LOOMBA (IIT RooRKEE)

FOR 100% SUCCESS IN JEE MAIN AND ADVANCED (IIT-JEE) ExAMPLE: Find all points of local maxima and local minima of the function f given by fx) 3x 3 f(x) 32 3 3 (r 1) (x1) (x) =0 at x = 1 and x =-1 Values of x to the right (say 1.1 etc.) to the left (say 0.9 etc.) to the right (say -0.9 etc.) Sign off'(x) 3(r 1) (r 1) >0 <0 <O >0 Close to 1 Close to -1 to the left (say -1.1 etc.) MATHEMATICS FOR IIT-JEE ER. VINEET LOOMBA (IIT RooRKEE)

FOR 100% SUCCESS IN JEE MAIN AND ADVANCED (IIT-JEE) Find the points at which the function f given by fx-(x-2f(x + 1 has i) local maxima Ex. (ii) local minima i) points of inflexion Sol.We have, 2 Now, f,(x)=0=> x=2,-1. Since (x 2(x +1) is always positive. So, sign of f(x) depends upon the sign of (x -2) (7x - 2). The changes in signs of f(x) as x increases through and 2 are shown in fig 2 Clearly, f(x) changes its sign from positive to negative as increases through 2/7 so, xis a point of local maximum. We observe that f(x) changes its sign from negative to positive as x increases through2 So, x 2 is a point of local minimum. There is no change in the sign of f(x) as increases through -1 So, x =-1 is a point of inflexion. MATHEMATICS FOR IIT-JEE ER. VINEET LOOMBA (IIT RooRKee)

FOR 100% SUCCESS IN JEE MAIN AND ADVANCED (IIT-JEE) NTH DeRIVATIVE TEST = 0 at x = a, then find dy', 0 at x = a, then neither maximum nor minimum at x = a. f 3 f0 at x-a, then find If > 0 i.e. positive at x a, then y is minimum at x a and 1 < 0 i.e. negative at x = a, then y is maximum at x = a and so on. MATHEMATICS FOR IIT-JEE ER. VINEET LOOMBA (IIT RooRKee)