1.

Verify rolles therom the function f(x)=(x-2)(x-6)in the interval(2,6)

Answer»

f(<klux>X</klux>)=x2+x−6,x∈[−3,<klux>2</klux>]

Since, f(x=x 2+x−6 is a polynomial and every polynomial function is continuous for all x∈R

So, f(x) is continuous at x∈[−3,2]

Also, every polynomial function is differentiable.

So, f(x) is differentiable at (−3,2)

Now,

f(−3)=9−3−6=0

f(2)=4+2−6=0

Hence, f(−3)=f(2)=0

Now, 

f′(x)=2x+1

f′(C)=2c+1

Since, all the conditions are satisfied,

[tex]f′(c)=0

2c+1=0

c=−21

And, −21∈[−3,2]

Hence, Rolle's theorem is verified.


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