Quadratics
- Find a quadratic polynomial which is \( 0 \) at \( x = 3 \),
is decreasing if \( x<1 \), and is increasing if \( x>1 \).
- Find a quadratic polynomial \( f \) which satisfies
\( f(0) = f'(0) = f''(0) = 2 \).
- Suppose that quadratic \( f \) has roots \( r \) and \( s \).
Show that \( f'(r) + f'(s) = 0 \).
- Show that the critical point of
a quadratic occurs midway between its roots.
Tags: critical-points, diff-at-a-point, diff, quadratic, computation, proof, a--g--a, a--a, t2, c1
This problem should be fairly straightforward for most students, and is a good way for them to get their hands on quadratics. If they’ve had the Mean Value Theorem, you might want to add an extra part, which generalizes part d: If f is a quadratic function of x, and if a and b are any real numbers, show that the value that satisfies the Mean Value Theorem for f in the interval (a,b) is the midpoint between a and b.
– Concha Gomez 1999-02-19