In a Math SL exam, how do you answer a "Show that" or "Demonstrate that"

[Piggy]

Hi, I'm trying to solve some exercises from my IB math book, but I'm not sure how I'm supposed to proceed with my answer so that I get the full mark. For example: "Show that y=2-7/(1+2x) has no horizontal tangents? What do they expect from me? 

[TheNintendoChip]

From what I've seen, they want you to logically deduce the answer. For these types of problems, you're given something that you essentially have to prove.

So what do they want from you here? Well first, you'd have to figure out what a horizontal tangent means. For one to exist, the derivative of the function in the domain must = 0. So what you're tying to show is that the derivative of y=2-7/(1+2x) with respect to x never equals 0, because if it did, there would be a horizontal tangent. You'd find the derivative, set it = 0 and show that there are no solutions, and hence there would be no horizontal tangent. Usually the final step in your answer will be the statement you're tying to prove.

I'll give you another example: Two vectors p and q are such that p=3i-3j+7k and q=-5i+2j+ak, and p is perpendicular to q. Show that a=3.

First, we know that because p and q are perpendicular, their scalar product is = 0. Thus, p . q =0. Next, we'd actually compute the scalar product: 3(-5)+-3(2)+7(a)=0. Finally, simplify: -15-6+7a=0 => -21+7a=0 => 7a=21 => a=3. We have shown what they asked, using deduction, and the question is complete!

I hope that clarifies everything