# MAT - Paper 2009 - Q3

> Source: https://ollybritton.com/notes/maths/problems/mat/2009/q3/ · Updated: 2021-10-13 · Tags: mat, notes, maths

## Flashcards
### 2021-10-13
##### What trick can you use for sketching $f(x)^2$??
Lightly sketch $f(x)$ and then make sure it's positive and increase all the gradients.

##### ![PHOTO POLYNOMIAL](polynomial.png) If this is $f(x)$, what does the graph of $f(x)^2$ look like??
![PHOTO POLYNOMIAL SQUARED](polynomial-squared.png)

##### We are given $$3n^2 - n - B + 3Bn \ge 4An^2 - An$$ for all $n > 1$. Equivalently $$(3 - 4A)n^2 + (A + 3B - 1)n - B \ge 0$$ Why can you show that $$3 - 4A \ge 0 \implies A \le 3/4$$ here??
Because if it $A \le 3/4$ there would be a negative for the $n^2$ term and so negative for large $n$.

##### $$-ax^2 + bx + c$$ where $a, b, c > 0$. What is ALWAYS true about this function for large values of $n$??
It is negative.

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