Q 1) Prove that f(x)=\frac{x^7}{7}-\frac{x^6}{6}+\frac{x^5}{5}-\frac{x^4}{4}+\frac{x^3}{3}-\frac{x^2}{2}+x-1 has exactly one real root
Q 2) In 1 hour a snail travels 60 meters. Prove that there was an Interval of 10 minutes where it traveled exactly 10 meters. (Not exactly Rolle's Theorem)
Q 3) Prove that f(x)=x3 - 3x + c never has both its roots in [0,1]
Q 4) if f(x)= ax3+bx2+cx+d, a≠0, Prove that f(x) cannot have more than one real roots, if b2<3ac
Q 5) Prove that there exists c in (a,b) such that f(c)f'(c)=c, given f is differentiable on (a,b) and f2(a)-f2(b)=a2-b2
Q 6) Let f is a continuous function on [a,b]
Prove that exists c in (a,b) such that \int_a^cf(x)\ \mathrm{dx}=(b-c)f(c)
Q 7) Prove LMVT using Rolle's Theorem
Q 8) Give an example to show that continuity in [a,b] and not just (a,b) is a necessary condition for both Rolle's and LMVT
Q 9) f and g are real functions continuous on [a,b] and differentiable on(a,b).
Show that there exists c in (a,b) such that f'(c)(b-c)+g'(c)(c-a)=(f(c)-f(a))+(g(b)-g(c))
Q 10) Let f , g continuous functions on [a , b] with f'(x)≠0 in (a , b).
Prove that there exists c in (a,b) such that : \frac{f'(c)}{f(a)-f(c)}+\frac{g'(c)}{g(b)-g(c)}= 1
62Q 11) Let f(x) continous in [a,b] satisfying the condition: \int^b_a f(x)dx=0
Show that there exists c in (a,b) such that:f(c)=2005 \int^c_a f(x)dx
Q 12) can we say that the result in 11 is true for any value or number other than 2005?
341msp your explanation for Q2 is not correct, because we need to show that there is an interval where the average speed was 1, not that the instantaneous velocity at a point is 1.
You will need a concept called the Universal Chord Theorem for this one
62Prophet sir, I think it can be done without that too...
just by constructing a function alone :)
Basically the function will use the idea of the proof :)
341that proof using the function u have in mind is the proof of the Universal Chord Theorem.
62yes i had never seen the universal chord theorem..
I just read it now after reading your post... and realized that the proof is the same as the function i had thought of :)
That is why i was wondering why i had not seen this kind of a problem when i was preparing!
3okie sir.Even i know that it is rong,i have given my contribution,so that i can get corrections from users.Thanq for correcting.
39what happened to post # 8 nishant bhaiyan ? u think its wrong???
shall i prove that??
1let us assume the given eqn has all roots real...so every equation we get after differentiating it will ahev real roots ....
1.differentiating given equation 5 times we get
15x2-5x+1=0
hence the roots are imaginary...so the given equations will have max possible even number of roots imaginary....hence only 1 real root