Date | May 2011 | Marks available | 3 | Reference code | 11M.1.hl.TZ1.12 |
Level | HL only | Paper | 1 | Time zone | TZ1 |
Command term | Sketch | Question number | 12 | Adapted from | N/A |
Question
Consider the function f(x)=lnxx , 0<x<e2 .
(i) Solve the equation f′(x)=0 .
(ii) Hence show the graph of f has a local maximum.
(iii) Write down the range of the function f .
Show that there is a point of inflexion on the graph and determine its coordinates.
Sketch the graph of y=f(x) , indicating clearly the asymptote, x-intercept and the local maximum.
Now consider the functions g(x)=ln|x|x and h(x)=ln|x||x| , where 0<x<e2 .
(i) Sketch the graph of y=g(x) .
(ii) Write down the range of g .
(iii) Find the values of x such that h(x)>g(x) .
Markscheme
(i) f′(x)=x1x−lnxx2 M1A1
=1−lnxx2
so f′(x)=0 when lnx=1, i.e. x=e A1
(ii) f′(x)>0 when x<e and f′(x)<0 when x>e R1
hence local maximum AG
Note: Accept argument using correct second derivative.
(iii) y⩽1e A1
[5 marks]
f″(x)=x2−1x−(1−lnx)2xx4 M1
=−x−2x+2xlnxx4
=−3+2lnxx3 A1
Note: May be seen in part (a).
f″(x)=0 (M1)
−3+2lnx=0
x=e32
since f″(x)<0 when x<e32 and f″(x)>0 when x>e32 R1
then point of inflexion (e32,32e32) A1
[5 marks]
A1A1A1
Note: Award A1 for the maximum and intercept, A1 for a vertical asymptote and A1 for shape (including turning concave up).
[3 marks]
(i) A1A1
Note: Award A1 for each correct branch.
(ii) all real values A1
(iii) (M1)(A1)
Note: Award (M1)(A1) for sketching the graph of h, ignoring any graph of g.
−e2<x<−1 (accept x<−1 ) A1
[6 marks]
Examiners report
Most candidates attempted parts (a), (b) and (c) and scored well, although many did not gain the reasoning marks for the justification of the existence of local maximum and inflexion point. The graph sketching was poorly done. A wide selection of range shapes were seen, in some cases showing little understanding of the relation between the derivatives of the function and its graph and difficulties with transformation of graphs. In some cases candidates sketched graphs consistent with their previous calculations but failed to label them properly.
Most candidates attempted parts (a), (b) and (c) and scored well, although many did not gain the reasoning marks for the justification of the existence of local maximum and inflexion point. The graph sketching was poorly done. A wide selection of range shapes were seen, in some cases showing little understanding of the relation between the derivatives of the function and its graph and difficulties with transformation of graphs. In some cases candidates sketched graphs consistent with their previous calculations but failed to label them properly.
Most candidates attempted parts (a), (b) and (c) and scored well, although many did not gain the reasoning marks for the justification of the existence of local maximum and inflexion point. The graph sketching was poorly done. A wide selection of range shapes were seen, in some cases showing little understanding of the relation between the derivatives of the function and its graph and difficulties with transformation of graphs. In some cases candidates sketched graphs consistent with their previous calculations but failed to label them properly.
Most candidates attempted parts (a), (b) and (c) and scored well, although many did not gain the reasoning marks for the justification of the existence of local maximum and inflexion point. The graph sketching was poorly done. A wide selection of range shapes were seen, in some cases showing little understanding of the relation between the derivatives of the function and its graph and difficulties with transformation of graphs. In some cases candidates sketched graphs consistent with their previous calculations but failed to label them properly.