DSE 2010: Q. 58 & 59

[Shivum]

How to solve these questions?

[Sinistral]

58)
a < b
f(a)=0 and f(b)=0 => there has to be atleast one minima/maxima in between (a,b). (ignoring the case wherein f(x)= 0 for all x belonging to [a,b])

f"(x)+f'(x) -1 =0
f'(x) has to be zero in between (a,b) because of the above inference.
So f"(x)-1=0 => f"(x) =1 => f"(x) >0 => f(x) has a minima in between (a,b)
it cant have a maxima since f"(x) is always greater than zero in between (a,b)
so option (b)

59)
once (58) is clear 59 is obvious.
if the graph of f(x) goes above x axis in between (a,b) then it has to come down to x axis at x=b (since f(b)=0 ). it wud lead to a local maxima in between (a,b). but we just saw that f(x) doesnt have a maxima in between (a,b). that means f(x) will never go above x axis in between (a,b). so option a

[Devika]

Sinistral,
could you explain Q60 from the same paper.
Thanks :)

[Sinistral]

on close observation it is obvious that g(y)= f inverse (y).
so g is also non decreasing because f inverse is non decreasing (as f in non decreasing)

the only problem we face is when f(x) becomes constant in part of its domain (it cant be constant throughout). because at those points inverse is not possible.
check from graph that if for a particular point on x axis say y1, if f(x1)=y1 such that 0<a<x1<b<1 and f is constant in between (a,b) . then g(y1) becomes = a. and g(y1+ϵ)>b . so g remains non decreasing.

similarly when 0<a<y1<b<1 and f is constant in between (a,b): let f(x')=y1. now g(y1) will always be x' ∀ 0<a<y1<b<1. here g behaves like a constant function.

basically seeing all these behaviour its obvious that g is always non decreasing

PS: all these point a,b,x1,y1,y etc are points on x axis.

[Do bats eat cats?]

Sorry to bring up an old thread, but how does one conclude that g = f inverse?

I'm trying to understand the question graphically and to me it looks like f and g are the same function, which doesn't make much sense, because then g defined on y wouldn't be a function at all.

[Arushi1]

Hey!

Can anyone please help me with question 60?

Thanks :)