What you are suggesting to me is just the verification of the result. Proving a result does not mean plugging the values and checking it.SteamKing said:You know that sine and cosine vary in amplitude between 0 and 1 on the interval in question. If you substitute the end values of the interval into the inequality, that should show that the inequality does not hold.
utkarshakash said:What you are suggesting to me is just the verification of the result. Proving a result does not mean plugging the values and checking it.
SteamKing said:If the inequality doesn't hold, then you are wasting time trying to prove that it does.
Verification that a relation holds at just one point in an interval is insufficient to prove that it holds at all points in the interval. However, showing that a relation fails to hold at just one point in an interval is sufficient to show that the relation does not hold.
jeppetrost said:Okay this is a bit messy, but it's the best I came up with. Say we define f(x)= 3sinx - xcosx, and we want to show f(x)<2x. Well, we see that at least the endpoints are all right [ f(0)=0 (okay since we don't actually consider 0), f(pi/2)=3 < pi ]. So if we can just show that the function never goes over the line 2x, we're done. Well, if we differentiate both sides, we can equivalently show that f '(x) < 2. Simply doing the differentiation gives us f '(x) = 2cosx-xsinx. We again check that the end points are okay: f '(0)= 2 (okay), f '(pi/2) = pi/2<2 (also okay). Now we need to show that f '(x) doesn't become greater than 2 (this is actually too strict, but we can anyway). So we calculate f ''(x) = x cos x-sinx and want to show that this is always negative. Well, at x=0 we have f ''(0) =0, so that's good. Now we go even further and find f '''(x) = -xsinx. This is obviously negative, which means that f ''(x) is also negative. This in turn means that f '(x) < 2, which means that f(x)<2x.
I hope you follow my train of thoughts. You just go deeper and deeper in the derivatives until you see something you know, and then go back up.
To prove an inequality using a graph, you will need to plot the two sides of the inequality on a coordinate plane. Then, analyze the graph to see where the two lines intersect, and determine if the inequality holds true for all values on the graph.
The purpose of proving an inequality via graph is to visually represent the relationship between the two sides of the inequality. This allows for a better understanding and visualization of the concept, and can also provide a more concrete proof.
Yes, there are some limitations to proving an inequality via graph. Graphs can only represent a limited number of values and may not accurately represent the entire range of values for the inequality. Also, some inequalities may be more difficult to graph, making it challenging to prove using this method.
In general, yes, you can use a graph to prove any type of inequality. However, some inequalities may be more difficult to graph and may require additional techniques or methods to prove.
Using a graph to prove an inequality can provide a visual representation of the concept, making it easier to understand and grasp. It also allows for a more concrete and visual proof, rather than relying solely on algebraic equations or proofs.