1 of 8 0-day streak

Analyse polynomial curves, stationary points and extrema

Question 1 of 8

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Consider the function with the displayed rule

Determine the coordinates of the stationary points

f(t)=\frac{1}{12}\,t^{3} - \frac{1}{8}\,t^{2} - \frac{1}{2}\,t - \frac{7}{24}

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

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
\(t\)	sign of f'(t)
\(-2\)	\(?\)
\(-1\)	\(?\)
\(\frac{1}{2}\)	\(?\)
\(2\)	\(?\)
\(3\)	\(?\)

sign of f'(t) at t=-2 sign of f'(t) at t=-1 sign of f'(t) at t=1/2 sign of f'(t) at t=2 sign of f'(t) at t=3

(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 \(t=2\)

The stationary point is a local maximum

The stationary point is a stationary point of inflection

The stationary point is a local minimum