A tangent to the graph of $y = \ln x$ passes through the origin.

(a)     Sketch the graphs of $y = \ln x$ and the tangent on the same set of axes, and hence find the equation of the tangent.

(b)     Use your sketch to explain why $\ln x \leqslant \frac{x}{{\text{e}}}$ for $x > 0$ .

(c)     Show that ${x^{\text{e}}} \leqslant {{\text{e}}^x}$ for $x > 0$ .

(d)     Determine which is larger, ${\pi ^{\text{e}}}$ or ${{\text{e}}^\pi }$ .

(a)

     A3

Note: Award A1 for each graph

A1 for the point of tangency.

 

point on curve and line is $(a,{\text{ }}\ln a)$     (M1)

$y = \ln (x)$

$\frac{{{\text{d}}y}}{{{\text{d}}x}} = \frac{1}{x} \Rightarrow \frac{{{\text{d}}y}}{{{\text{d}}x}} = \frac{1}{a}\,\,\,\,\,{\text{(when }}x = a)$     (M1)A1

EITHER

gradient of line, m, through (0, 0) and $(a,{\text{ }}\ln a)$ is $\frac{{\ln a}}{a}$     (M1)A1

$\Rightarrow \frac{{\ln a}}{a} = \frac{1}{a} \Rightarrow \ln a = 1 \Rightarrow a = {\text{e}} \Rightarrow m = \frac{1}{{\text{e}}}$     M1A1

OR

$y - \ln a = \frac{1}{a}(x - a)$     (M1)A1

passes through 0 if

$\ln a - 1 = 0$     M1

$a = e \Rightarrow m = \frac{1}{{\text{e}}}$     A1

THEN

$\therefore y = \frac{1}{{\text{e}}}x$     A1

[11 marks]

 

(b)     the graph of $\ln x$ never goes above the graph of $y = \frac{1}{{\text{e}}}x$ , hence $\ln x \leqslant \frac{x}{{\text{e}}}$     R1

[1 mark]

 

(c)     $\ln x \leqslant \frac{x}{{\text{e}}} \Rightarrow {\text{e}}\ln x \leqslant x \Rightarrow \ln {x^{\text{e}}} \leqslant x$     M1A1

exponentiate both sides of $\ln {x^{\text{e}}} \leqslant x \Rightarrow {x^{\text{e}}} \leqslant {{\text{e}}^x}$     R1AG

[3 marks]

 

(d)     equality holds when $x = {\text{e}}$     R1

letting $x = \pi \Rightarrow {\pi ^{\text{e}}} < {{\text{e}}^\pi }$     A1     N0

[2 marks]

Total [17 marks]

This was the least accessible question in the entire paper, with very few candidates achieving high marks. Sketches were generally done poorly, and candidates failed to label the point of intersection. A ‘dummy’ variable was seldom used in part (a), hence in most cases it was not possible to get more than 3 marks. There was a lot of good guesswork as to the coordinates of the point of intersection, but no reasoning showed. Many candidates started with the conclusion in part (c). In part (d) most candidates did not distinguish between the inequality and strict inequality.