1 of 8 0-day streak

Locate a polynomial turning point numerically

Question 1 of 8

Calculator allowed

Use a table of values to locate the requested stationary point

P(x)=-\frac{1}{4}x^{4} - \frac{3}{2}x^{3} + \frac{9}{8}x^{2} + \frac{81}{8}x + \frac{2549}{192}

The model output is \(P(x)\) for inputs in the meaningful domain \(x\in[-1, 4]\)

Graph of the polynomial over the stated model domain

A preliminary numerical check has narrowed the initial window before the displayed table. Complete the table in order and give both coordinates to 1 decimal place

(a)1 mark

Which interval and extremum type are consistent with the polynomial and the first table?

Use \(x\in[-2, 3]\). The stationary point is \(a local maximum\)

Use \(x\in[0, 5]\). The stationary point is \(a local minimum\)

Use \(x\in[-1, 4]\). The stationary point is \(a local minimum\)

Use \(x\in[-1, 4]\). The stationary point is \(a local maximum\)

(b)3 marks

Complete the coarse table. The input step is \(\frac{1}{5}\); leave the supplied entries unchanged and enter the missing exact values

input	output
\(\frac{11}{10}\)	\(\frac{43898}{1875}\)
\(\frac{13}{10}\)	\(\frac{182477}{7500}\)
\(\frac{3}{2}\)	?
\(\frac{17}{10}\)	?
\(\frac{19}{10}\)	?

\(P(\frac{3}{2})\) \(P(\frac{17}{10})\) \(P(\frac{19}{10})\)

Give the approximate horizontal coordinate of the maximum to 1 decimal place

horizontal coordinate \(x\)

Format: 1 d.p.

(d)1 mark

Give the corresponding approximate output value to 1 decimal place

corresponding output \(P(x)\)

Format: 1 d.p.

(e)1 mark

Which interpretation is supported by the refined table and the stated domain?

The model output has a local minimum near x=1.5; the model output there is about 24.7 units

The model output has a local maximum near x=1.5; the model output there is about 24.7 units

The first table gives its maximum at x=1.5; its model output is about 25.7 units

The model output has a local maximum near x=24.7; the model output there is about 1.5 units