Question

Given that f'(x)= product of (n-x)^n from n=1 to 101.
No. Of relative maximum of f(x) is

Answer 1

@kainui

Answer 2

Product is like (1-x)(2-x)^2(3-x)^3 ....and so on

Answer 3

@hartnn

Answer 4

@parthkohli

Answer 5

Only studied this in physics.

1. \(f'(x_0)=0\)
2. \(f''(x_0) < 0\)

Answer 6

Yep it is this way only in maths too...
This is second derivative test though..

Answer 7

Yeah, what else are we supposed to do?

Answer 8

First derivative says that if f'(x) changes sign from positive to negative then there occurs a maxima
And reverse for minima

Answer 9

@michele_laino

Answer 10

I think that the sign of the first derivative is the sign of this function:
\[g\left( x \right) = \left( {1 - x} \right)\left( {2 - x} \right)\left( {3 - x} \right)...\left( {101 - x} \right)\]

Answer 11

Sir how ?
Actually we can do one thing
Since (2-x)^2 is all time positive similarly such terms like (4-x)^4 is also +ve

Answer 12

So we can ignore them ..
Isn't it?

Answer 13

Then we will be left with only odd term which again will include just
(1-x)(3-x)(5-x)...and so on.

Answer 14

sorry I have made a typo, there are odd numbers only:
\[g\left( x \right) = \left( {1 - x} \right)\left( {3 - x} \right)\left( {5 - x} \right)\left( {7 - x} \right)...\left( {101 - x} \right)\]

Answer 15

Yes ...sir it has to be like this now...

Answer 16

\(g(x)\) is positive when \(all\) factors are positive, since we have an \(odd\) number of factors

Answer 17

or when there are an \(even\) number of negative factors

Answer 18

BT sir there are even no. of factors

Answer 19

From 1 to 101 we have even no. of terms in this case....

Answer 20

yes! you are right! sorry again!

Answer 21

No problem...

Answer 22

Sir what next?

Answer 23

If we have even no of terms then we can say that product will be positive when they are pairwise positive or negative

Answer 24

for example if we consider these three factors:
(I'm applying the principle of Mathematical induction)
\[g\left( x \right) = \left( {1 - x} \right)\left( {3 - x} \right)\left( {5 - x} \right)\]

Answer 25

we have this diagram:

where the sign refer to the sign of product of \((1-x)(3-x)(5-x)\)

Answer 26

as we can see only he points \(x=1,\;x=5\) are maximum for \(g(x)\)

Answer 27

the*

Answer 28

so, by means of the principle of Mathematical induction, wehave the maxi8mum points at these ponts:
\(x=1,5,9,13,...\)
namely the points of maximum for \(f(x)\) are aan arithmetic sequence, whose first term is \(1\), and the constant is \(4\), and the last term is \(x=101\)
so we have:
\[101 = 1 + \left( {n - 1} \right) \times 4\]
from such equation, I get \(n=26\) points of maximum

Answer 29

Thank you so much sir!!!