## 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 JEE MAIN AND ADVANCED 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) Monotonicity at a point: (a) A uction fo) is called an increasing function at point x a, if in a suffciently small neighbourhood of x- a fa+h) f(a) f(a-h) fa-h) a-h a a+h (b) A unction fo) is called a decreasing function at point x - a if in a sufficiently small neighbourhood of x a fa -h)> fa)> fa+ h) fa-h) f(a) fa-h) fa+h) f(a) a+h) a-h a a+h xra MATHEMATICS FOR IIT-JEE ER. VINEET LOOMBA (IIT RooRKEE)

FOR 100% SUCCESS IN JEE MAIN AND ADVANCED (IIT-JEE) Note: If x = a is a boundary point then use the appropriate one sides inequality to test Monotonicity of fx). f(a) f(a+h) f(a-h) fla) xra f(a)> fa-h) inceasing at x =a fath)> fa) decreasing at x=a MATHEMATICS FOR IIT-JEE ER. VINEET LOOMBA (IIT RooRKEE)

FOR 100% SUCCESS IN JEE MAIN AND ADVANCED (IIT-JEE) Strictly increasing function A finction/to)is said to be a strictly increasing fiunction on (a,b)if x, < x, f(x) <f(x) for all x" x, e (a, b) Thus,f(x) is strictly increasing on (a, b) ifthe values off(x) increase with the increase in the values of x. Graphically, f(a) is increasing on (a, b) ifthe graphy (x) moves up as x moves to the right. The graph in Fig. 1 is the graph ofa strictly increasing function on (a.b). flx?) 01ax1 x2 b MATHEMATICS FOR IIT-JEE ER. VINEET LOOMBA (IIT RooRKEE)

FOR 100% SUCCESS IN JEE MAIN AND ADVANCED (IIT-JEE) Strictly decreasing function A functionf(x) is said to be a strictly decreasing function on (a, b) if Xi < X2 f(x;) >f(x2) for all xi, X2 (a, b) Thus, f(x) is strictly decreasing on (a, b) if the values of f (x) de- crease with the increase in the values ofx. Graphically it means that f(x) is a decreasing function on (a, b) if its graph moves down as x moves to the right. The graph in Fig. 2 is O2 the graph ofa strictly decreasing function. MATHEMATICS FOR IIT-JEE ER. VINEET LOOMBA (IIT RooRKEE)

FOR 100% SUCCESS IN JEE MAIN AND ADVANCED (IIT-JEE) Monotonically increasing (MI) / Monotonically Decreasing (MD) (i) fis strictly increasing in (a, b) if f'(x) > 0 for each xe (a, b) (ii) f is strictly decreasing in (a, b) if f'(x) < 0 for each x e (a, b) (a) A function fx) is said to be increasing in (a, b) if f(x)20 (b) f(x) is said to be decreasing in (a, b) if f(x) S 0 MATHEMATICS FOR IIT-JEE ER. VINEET LOOMBA (IIT RooRKEE)

FOR 100% SUCCESS IN JEE MAIN AND ADVANCED (IIT-JEE) Example Show that the function f given by f(x) =x3-3x2 + 4x, xe R is strictly increasing on R. Solution Note that f'(x) = 3x2 _ 6x + 4 t)y 3(x2- 2x1) 1 -30x 1>0, in every interval of R Therefore, the function fis strictly increasing on R MATHEMATICS FOR IIT-JEE ER. VINEET LOOMBA (IIT RooRKEE)

FOR 100% SUCCESS IN JEE MAIN AND ADVANCED (IIT-JEE) Illustration The function (logx increases on the intervwal log.r (D) none of these SOLUTION: (C) Clearly,f(x) is defined for>0 Now, log r -1 (log x) f(r)- MATHEMATICS FOR IIT-JEE ER. VINEET LOOMBA (IIT RooRKEE)

FOR 100% SUCCESS IN 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. * Share among your peers as SHARING is CARING!!