1 of 8 0-day streak

Analyse polynomial curves, stationary points and extrema

Question 1 of 8

No calculator

The rule for a function is shown below

Obtain the stationary coordinates of the curve

p(u)=\frac{1}{6}\,u^{3} + \frac{3}{8}\,u^{2} + \frac{1}{4}\,u + \frac{1}{3}

The domain is all real values of \(u\)

Report derivative-sign intervals as open intervals split at every derivative zero, even when the sign does not change. Do not include stationary inputs in those reported intervals

(a)5 marks

Complete the derivative-sign table. Enter -1 for a negative derivative, 0 for zero and 1 for a positive derivative

Derivative sign table
\(u\)	sign of p'(u)
\(-2\)	\(?\)
\(-1\)	\(?\)
\(-\frac{3}{4}\)	\(?\)
\(-\frac{1}{2}\)	\(?\)
\(\frac{1}{2}\)	\(?\)

sign of p'(u) at u=-2 sign of p'(u) at u=-1 sign of p'(u) at u=-3/4 sign of p'(u) at u=-1/2 sign of p'(u) at u=1/2

(b)5 marks

Complete the exact feature table with the y-intercept and stationary coordinates

Function features
Feature	Value
y-intercept ordinate	\(?\)
stationary input 1	\(?\)
stationary ordinate 1	\(?\)
stationary input 2	\(?\)
stationary ordinate 2	\(?\)

y-intercept ordinate stationary input 1 stationary ordinate 1 stationary input 2 stationary ordinate 2

Classify the stationary point at \(u=-\frac{1}{2}\)

The stationary point is a local minimum

The stationary point is a stationary point of inflection

The stationary point is a local maximum