IB DP Maths Topic 6.2 The second derivative SL Paper 2

 

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Question

The following diagram shows the graph of \(f(x) = {{\rm{e}}^{ – {x^2}}}\) .


The points A, B, C, D and E lie on the graph of f . Two of these are points of inflexion.

Identify the two points of inflexion.

[2]
a.

(i)     Find \(f'(x)\) .

(ii)    Show that \(f”(x) = (4{x^2} – 2){{\rm{e}}^{ – {x^2}}}\) .

[5]
b(i) and (ii).

Find the x-coordinate of each point of inflexion.

[4]
c.

Use the second derivative to show that one of these points is a point of inflexion.

[4]
d.
Answer/Explanation

Markscheme

B, D     A1A1     N2

[2 marks]

a.

(i) \(f'(x) = – 2x{{\rm{e}}^{ – {x^2}}}\)     A1A1     N2

Note: Award A1 for \({{\rm{e}}^{ – {x^2}}}\) and A1 for \( – 2x\) .

(ii) finding the derivative of \( – 2x\) , i.e. \( – 2\)     (A1)

evidence of choosing the product rule     (M1)

e.g. \( – 2{{\rm{e}}^{ – {x^2}}}\) \( – 2x \times – 2x{{\rm{e}}^{ – {x^2}}}\)

\( – 2{{\rm{e}}^{ – {x^2}}} + 4{x^2}{{\rm{e}}^{ – {x^2}}}\)     A1

\(f”(x) = (4{x^2} – 2){{\rm{e}}^{ – {x^2}}}\)     AG     N0

[5 marks]

b(i) and (ii).

valid reasoning     R1

e.g. \(f”(x) = 0\)

attempting to solve the equation     (M1)

e.g. \((4{x^2} – 2) = 0\) , sketch of \(f”(x)\)

\(p = 0.707\) \(\left( { = \frac{1}{{\sqrt 2 }}} \right)\) , \(q = – 0.707\) \(\left( { = – \frac{1}{{\sqrt 2 }}} \right)\)     A1A1     N3

[4 marks]

c.

evidence of using second derivative to test values on either side of POI     M1

e.g. finding values, reference to graph of \(f”\) , sign table

correct working     A1A1

e.g. finding any two correct values either side of POI,

checking sign of \(f”\) on either side of POI

reference to sign change of \(f”(x)\)     R1     N0

[4 marks]

d.
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