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## Homework Statement

I failed my math exam last year (first year of college) because I basically had no math classes during my last two years of high school. So I would like to learn the specific materials to be able to solve these problems.

So I guess this isn't a traditional question, but I do have some question examples from the exam, I have absolutely no idea how to go about answering them, but I'm supposed to know.

I was thinking of using online resources to catch up (maybe Khan Academy?) and to be ultimately able to do this.

**2. Relevant questions**

- Let V be a real vectorial space of dimension 3. Let's take a non free minimal part Y of V (ie. Y is not free but the retraction of whatever vector from Y gives a free part) a) Show that such a part has maximum 4 vectors b) What are all the cardinals of such parts?
- Give the Cartesian equation perpendicular to the right of the equations of the plan ℝ³

x = k+2

y = 2k+3

z = 3k+4 - Let
*f*: ]0,1[ → ℝ be a derivable function. Let us suppose that exactly two points of ]0,1[ give us a minimum of*f*, and that these two minimums in question are strict locals (without necessarily being equals). In this case, the number of points of ]0,1[ that give a strict local maximum of*f*can only take a few values. Give the set of these values and justify. - Let us consider the function ℝ² → ℝ : (x,y) ↦ 2x² + 3y² + 5x + 7y. For [tex]\overrightarrow{v} = (1,1)[/tex] and [tex]\overrightarrow{w} = (1,2)[/tex] two vectors linked by (0,0), calculate the first-order partial derivatives and directional derivatives.

## The Attempt at a Solution

That's it. And by no means am I asking anyone to solve them for me, not that. Like I said, I haven't the slightest clue how to solve these. So if you could just like tell me the names of the subjects I should be learning to be able to answer these in the future so that I can get started on it, it would mean the world to me.

I've looked around and this seemed the most appropriate part of the forum to post this, if it isn't then please tell me so that I can fix it.