MAT - Paper 2009 - Q3
Flashcards
Sketching the square of a function
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.
If this is $f(x)$, what does the graph of $f(x)^2$ look like?
Bounding a quadratic in n
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.