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Date May 2008 Marks available 7 Reference code 08M.2.hl.TZ2.6
Level HL only Paper 2 Time zone TZ2
Command term Find and Justify Question number 6 Adapted from N/A

Question

Consider the curve with equation f(x)=e2x2 for x<0 .

Find the coordinates of the point of inflexion and justify that it is a point of inflexion.

Markscheme

METHOD 1

EITHER

Using the graph of y=f(x)     (M1)

     A1

 

The maximum of f(x) occurs at x = −0.5 .     A1

OR

Using the graph of y=f.     (M1)

     A1

 

The zero of f''(x) occurs at x = −0.5 .     A1

THEN

Note: Do not award this A1 for stating x = ±0.5 as the final answer for x .

 

f( - 0.5) = 0.607( = {{\text{e}}^{ - 0.5}})     A2

Note: Do not award this A1 for also stating (0.5, 0.607) as a coordinate.

 

EITHER

Correctly labelled graph of f'(x) for x < 0 denoting the maximum f'(x)     R1

(e.g. f'( - 0.6) = 1.17 and f'( - 0.4) = 1.16 stated)     A1     N2

OR

Correctly labelled graph of f''(x) for x < 0 denoting the maximum f'(x)     R1

(e.g. f''( - 0.6) = 0.857 and f''( - 0.4) = - 1.05 stated)     A1     N2

OR

f'(0.5) \approx 1.21. f'(x) < 1.21 just to the left of x = - \frac{1}{2}

and f'(x) < 1.21 just to the right of x = - \frac{1}{2}     R1

(e.g. f'( - 0.6) = 1.17 and f'( - 0.4) = 1.16 stated)     A1     N2

OR

f''(x) > 0 just to the left of x = - \frac{1}{2} and f''(x) < 0 just to the right of x = - \frac{1}{2}     R1

(e.g. f''( - 0.6) = 0.857 and f''( - 0.4) = - 1.05 stated)     A1     N2

[7 marks] 

METHOD 2

f'(x) = - 4x{{\text{e}}^{ - 2{x^2}}}     A1

f''(x) = - 4{{\text{e}}^{ - 2{x^2}}} + 16{x^2}{{\text{e}}^{ - 2{x^2}}}\,\,\,\,\,\left( { = (16{x^2} - 4){{\text{e}}^{ - 2{x^2}}}} \right)     A1

Attempting to solve f''(x) = 0     (M1)

x = - \frac{1}{2}     A1

Note: Do not award this A1 for stating x = \pm \frac{1}{2} as the final answer for x .

 

f\left( { - \frac{1}{2}} \right) = \frac{1}{{\sqrt {\text{e}} }}{\text{ }}( = 0.607)     A1

Note: Do not award this A1 for also stating \left( {\frac{1}{2},\frac{1}{{\sqrt {\text{e}} }}} \right) as a coordinate.

 

EITHER

Correctly labelled graph of f'(x) for x < 0 denoting the maximum f'(x)     R1

(e.g. f'( - 0.6) = 1.17 and f'( - 0.4) = 1.16 stated)     A1     N2

OR

Correctly labelled graph of f''(x) for x < 0 denoting the maximum f'(x)     R1

(e.g. f''( - 0.6) = 0.857 and f''( - 0.4) = - 1.05 stated)     A1     N2

OR

f'(0.5) \approx 1.21. f'(x) < 1.21 just to the left of x = - \frac{1}{2}

and f'(x) < 1.21 just to the right of x = - \frac{1}{2}     R1

(e.g. f'( - 0.6) = 1.17 and f'( - 0.4) = 1.16 stated)     A1     N2

OR

f''(x) > 0 just to the left of x = - \frac{1}{2} and f''(x) < 0 just to the right of x = - \frac{1}{2}     R1

(e.g. f''( - 0.6) = 0.857 and f''( - 0.4) = - 1.05 stated)     A1     N2

[7 marks]

Examiners report

Most candidates adopted an algebraic approach rather than a graphical approach. Most candidates found f'(x) correctly, however when attempting to find f''(x), a surprisingly large number either made algebraic errors using the product rule or seemingly used an incorrect form of the product rule. A large number ignored the domain restriction and either expressed x = \pm \frac{1}{2} as the x-coordinates of the point of inflection or identified x = \frac{1}{2} rather than x = - \frac{1}{2}. Most candidates were unsuccessful in their attempts to justify the existence of the point of inflection.

Syllabus sections

Topic 6 - Core: Calculus » 6.3 » Points of inflexion with zero and non-zero gradients.

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