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Worked solution

Match slope and concavity intervals on a cubic graph

Using the given graph of y=0.1x(x−6)(x+6)y=0.1x(x-6)(x+6), match (a) strictly increasing, (b) strictly decreasing, (c) concave upwards and (d) concave downwards to the printed sets: {x∈R:−3.46<x<3.46}\{x\in\mathbb R:-3.46<x<3.46\}; {x∈R:x<0}\{x\in\mathbb R:x<0\}; {x∈R:x>0}\{x\in\mathbb R:x>0\}; {x∈R:x<−3.46 or x>3.46}\{x\in\mathbb R:x<-3.46\text{ or }x>3.46\}.

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Graph of 0.1x(x−6)(x+6), roots −6,0,6, printed approximate turning labels (−3.46,8.31),(3.46,−8.31). Concavity changes at 0; turning abscissae exact ±2sqrt3.
Graph of 0.1x(x−6)(x+6), roots −6,0,6, printed approximate turning labels (−3.46,8.31),(3.46,−8.31). Concavity changes at 0; turning abscissae exact ±2sqrt3.
Graph of 0.1x(x−6)(x+6), roots −6,0,6, printed approximate turning labels (−3.46,8.31),(3.46,−8.31). Concavity changes at 0; turning abscissae exact ±2sqrt3.

CJC · Lecture · Example 4

(a) Strictly increasing

Match to the fourth printed set: x<−3.46x<-3.46 or x>3.46x>3.46, using the graph’s rounded labels. The Teacher arrow connects (a) to this last choice. Independently, y′=0.3x2−3.6y^{\prime}=0.3x^2-3.6 is positive for x<−23x<-2\sqrt3 or x>23x>2\sqrt3. These are the exact open intervals; 3.46 is a rounded boundary, not an exact derivative root.

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(b) Strictly decreasing

Match to the first printed set: −3.46<x<3.46-3.46<x<3.46 as displayed on the source graph. The exact derivative-sign interval is −23<x<23-2\sqrt3<x<2\sqrt3. The Teacher arrow links (b) to the first choice. Do not turn the rounded bound into an exact assertion about inputs between 3.46 and 232\sqrt3.

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(c) Concave upwards

Match to the third printed set: x>0x>0. The Teacher arrow is horizontal from (c) to this choice. The independent check y′′=0.6x>0y^{\prime\prime}=0.6x>0 agrees.

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(d) Concave downwards

Match to the second printed set: x<0x<0, following the Teacher arrow. Here y′′=0.6x<0y^{\prime\prime}=0.6x<0. The printed turning coordinates are (−3.46,8.31)(-3.46,8.31) and (3.46,−8.31)(3.46,-8.31); exact values are (−23,4.83)(-2\sqrt3,4.8\sqrt3) and (23,−4.83)(2\sqrt3,-4.8\sqrt3). The original graph also marks x-intercepts −6,0,6-6,0,6.

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Common mistakes

The sets are shuffled matching choices. Slope and concavity use different derivatives; graph decimals are rounded.

Solution notes

Both originals visually compared. Complete source working and distinct Teacher guidance integrated; independently authored domain/precision qualifications are identified. No printed marks or past-paper attribution.

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