Inequality involving Calculus - Qs (1 Viewer)

[a1079atw]

Hi everyone,

Can someone please show me how to do these two questions? Don't really know where to start :s


Thanks in advance!

 

[Kurosaki]

Hi everyone,

Can someone please show me how to do these two questions? Don't really know where to start :s
View attachment 30366
View attachment 30367
Thanks in advance!

For the first one, let f(x)=
Differentiate that. Show that f(x) is an increasing function (i.e. derivative > 0). find f(0). Since f(0)= blah and f'(x)>0, then what conclusions can you draw?

I can do the second one but not sure how to express it :S. Do you want me to have a go anyway?

 

[dunjaaa]

As for the first one, the best way to prove these type of questions is by moving it all on one side and calling it f(x) and proving > or < 0. This can be done by inspecting stationary points. There always tends to be one minimum/maximum stationary point with y-ordinate 0. Hence, you can deduce from there onwards.

As for the second one, work with the inequality given. If 0 < t < 1, 1 < t+1 < 2, when you flip everything the inequality signs also flip and you get your required result. Then part 2 is just integrating everything between t=0 to t=u and the result falls out immediately

 

[Squar3root]

basic steps

move everything to one side so it equals zero

the trig functions can be combined using the auxiliary method (i got cos(x+75)) [there are other answers]

plot the graph and hence true for all x > 0

EDIT: didn't realise that there were 2 questions but the solution is below. IMO, sy123's solution is very long for the first bit; but still valid

 

[Sy123]

 

[seanieg89]

basic steps

move everything to one side so it equals zero

the trig functions can be combined using the auxiliary method (i got cos(x+75)) [there are other answers]

plot the graph and hence true for all x > 0

EDIT: didn't realise that there were 2 questions but the solution is below. IMO, sy123's solution is very long for the first bit; but still valid

Care to elaborate on your own proof? What are you saying is equal to cos(x+75)?

Perhaps I am missing something basic, but I do not see how this can be done by a simple rearrangement and graph.

 

[Squar3root]

Care to elaborate on your own proof? What are you saying is equal to cos(x+75)?

Perhaps I am missing something basic, but I do not see how this can be done by a simple rearrangement and graph.

oops my bad. i forgot to multiply the cos x with x auxiliary method cannot be used. my solution is invalid

here's the graph though:

 

[seanieg89]

oops my bad. i forgot to multiply the cos x with x auxiliary method cannot be used. my solution is invalid

here's the graph though:

Well I knew what the graph looks like, my solution was similar to Sy's, and you need to do something like that to come up with the graph.

 

[Squar3root]

Well I knew what the graph looks like, my solution was similar to Sy's, and you need to do something like that to come up with the graph.

maple?

 

[seanieg89]

Well obviously a computer can do it, or you could ask someone else as well.

I meant that any method involving "graphing" requires work like Sy's to justify rigorously in exam conditions.

 

[a1079atw]

Thank you all, I understand now