Question

if y=3x+2cos x, find dy/dx and hence show that y increases as x increases.

i know how to get dy/dx but i dont know how to prove that 'y increases as x increases' please help!

Answer 1

You will have found then that dy/dx=3-2sinx.
Take a good look at this formula.
What are the possible outcomes (the range) of dy/dx?

Answer 2

it is pi/6 or - pi/6
i think?!?!?

Answer 3

No, you should remember that you can put any number x into dy/dx=3-2sinx.
If you do this, what are all the possible outcomes of 3-2sinx?
Maybe first: what are the possible outcomes of sinx?

Answer 4

lowest x value to higher one?

Answer 5

You know the range of sinx: it is [-1,1]. Whatever x to put into the sine function, the outcome is always a number between -1 and 1 (included).
So if you calculate 2sinx, the outcome of that is always a number in the interval [-2,2].
Are you with me so far?

Answer 6

ahhh true..

Answer 7

i forgot about the restrictions

Answer 8

OK, now we have this number in the interval [-2,2].
To calculate dy/dx, you subtract the number from 3, after all, dy/dx=3-2sinx.
To see what the possible outcomes of dy/dx are, look at the extremes -2 and 2:
3--2=5
3-2=1

So...dy/dx alis always in the interval [1,5].
That means: dy/dx is always positive.

Answer 9

so.. 2sinx could be replaced with -2 or 2?

Answer 10

Also, you know that dy/dx is the slope of the tangent line to the graph of y=3x+2cosx.
So the slope is always a positive number, so y is always increasing.

Remember:
y' positive means y increasing
y' negative means y decreasing

Answer 11

No you cannot replace 2sinx by 2 or -2, it is just that I want to see what values 3-2sinx can take.

At the very LEAST it gets to 3-2 = 1
At the MOST it gets to 3--2 = 5, so the range is [1,5]

Answer 12

Your statement is very easy! thank you so much for the help!

Answer 13

Happy New Year!

Answer 14

you too!! :)