Cubics
- Find the maximum and minimum values of
\( f(x) = x^{3}-3x^{2}+3x+4 \) on \( [0,2] \).
- Find the highest and lowest points on the graph of
\( f(x) = x^{3}-3x+6 \) on
the following intervals:
(i) \( [-2,2] \); (ii) \( [-2,3] \); and (iii) \( (-2,3) \).
- Sketch a graph of \( f(x) = x^{3}-3x+6 \)
indicating local maxima, minima, and points of inflection.
- Find the equation of the cubic
for which the origin is a point of
inflection, and \( (-2,16) \) are the coordinates
of the local maximum point.
- Show that the maximum and minimum values of the function
\( f(x) = x^{3}+ax^{2}+bx+c \) on the interval \( [p,q] \)
occur at the endpoints if \( a^{2} <3b \).
Tags: critical-points, diff-at-a-point, curve-sketching, diff, cubic, computation, sketching, proof, a--ag, t2, c2
This is a good problem to emphasize the differences between absolute and local extrema, and to make sure they are checking endpoints of closed intervals.
– Concha Gomez 1999-02-19