Sketch the graph of the function f(x)=−x3+3x2–7. List the coordinated of where extrema or points of inflection occur. State where the function is increasing or decreasing as well as where it is concave up or concave down.

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Sketch the graph of the function f(x)=−x3+3x2–7. List the coordinated of where extrema or points of inflection occur.  State where the function is increasing or decreasing as well as where it is concave up or concave down.

BleabyinfibiaG · Accepted Answer

f(x)=−x3+3x2–7 a) Extrema point: f(x)=0 f1(x)=ddx(−x3+3x2–7)=−3x2+6x −3x2+6x=0 x(−3x+6)=0 x=0,2 So, at x=2,y=−8+3×4–7=−3 Maxima (2,−3) At x=0,y=−0+0–7=−7 Minima (0,−7) b) Inflection point: f(x)=0 f2(x)=ddx(−3x2+6x) −6x+6=0 x=1, so inflection point occurs at (1,−5) c)Increasing, when f1(x)&amp;gt;0 −3x2+6x&amp;gt;0 x∈(0,2) Decreasing, when f1(x)&amp;lt;0 −3x2+6x&amp;lt;0 x∈(−∞,0)∪(2,∞) d) Concave up: f2(x)&amp;gt;0 −6x+6&amp;gt;0 x∈(−∞,1) Concave down: f2(x)&amp;gt;0 x∈(1,∞)

Sketch the graph of the function f(x) = - x^3 + 3x^2 – 7. List the coordinated of where extrema or points of inflection occur. State where the function is increasing or decreasing as well as where it is concave up or concave down.

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