Proving with mean value theorem

AI Thread Summary
The discussion revolves around applying the Mean Value Theorem (MVT) to prove that |g(a+h)| < Mh^2 under the given conditions. It begins by noting that since g(a) = g'(a) = 0, the MVT can be applied to g' on the interval [a, a+k], leading to the conclusion that g'(k) = g''(c) for some c in (a, a+k). Given that |g''(x)| < M, it follows that |g'(k)| < Mh. Subsequently, integrating this result over the interval [0, h] leads to the final inequality |g(a+h)| < Mh^2. This proof effectively demonstrates the relationship between the second derivative and the behavior of the function over the specified interval.
trap
Messages
51
Reaction score
0
Suppose that g(a) = g'(a)=0 and |g''(x)| < M for all x in [a, a+h] (for some positive constant M). Show that |g(a+h)| < Mh^2.
(Hint: Let k be any number such that 0<= k <= h and apply Mean Value Theorem to g' on [a,a+k].)
 
Physics news on Phys.org
Well, what does the mean value theorem say, and what does it imply when you use it as hinted?
 
I'd also keep in mind the definition of a derivative.

--J
 
Thread 'Collision of a bullet on a rod-string system: query'

Thread 'Collision of a bullet on a rod-string system: query'

In this question, I have a question. I am NOT trying to solve it, but it is just a conceptual question. Consider the point on the rod, which connects the string and the rod. My question: just before and after the collision, is ANGULAR momentum CONSERVED about this point? Lets call the point which connects the string and rod as P. Why am I asking this? : it is clear from the scenario that the point of concern, which connects the string and the rod, moves in a circular path due to the string...
View full post »
Thread 'A cylinder connected to a hanged mass'

Thread 'A cylinder connected to a hanged mass'

Let's declare that for the cylinder, mass = M = 10 kg Radius = R = 4 m For the wall and the floor, Friction coeff = ##\mu## = 0.5 For the hanging mass, mass = m = 11 kg First, we divide the force according to their respective plane (x and y thing, correct me if I'm wrong) and according to which, cylinder or the hanging mass, they're working on. Force on the hanging mass $$mg - T = ma$$ Force(Cylinder) on y $$N_f + f_w - Mg = 0$$ Force(Cylinder) on x $$T + f_f - N_w = Ma$$ There's also...
View full post »

Similar threads

Spring Problem Involving Variables and Constants Only
Replies
3
Views
1K
Initial velocity of bird landing on horizontal wire modeled as spring
Replies
7
Views
912
A block dropping onto a spring system
2
Replies
58
Views
2K
Projectile motion with ##N## bounces on the ground
Replies
21
Views
947
Pressure of water coming out of a hose nozzle
Replies
7
Views
934
Two balls, dropped with a delay of ##\Delta t##, meet after rebound
Replies
34
Views
2K
Height of a stable droplet on a perfectly wetting surface
2
Replies
63
Views
4K
Ball launched from a height of 5 meters --- Projectile motion
Replies
12
Views
998
Work Done in slowly pulling a thread
Replies
28
Views
1K
The energy of a ball shot upwards
Replies
5
Views
1K

Hot Threads

Recent Insights

Back
Top