Decreasing curves and the largest tangent intercept

Original exponential curveSource-faithful qualitative diagram; distances are schematic unless an exact function is stated.xyOy=xe^(−x)
Question diagram 1
Original exponential curveSource-faithful qualitative diagram; distances are schematic unless an exact function is stated.xyOy=xe^(−x)

Given graph y=xe^(−x), including the negative-x branch.

The diagram shows a sketch of the curve y=xe−xy=xe^{-x}.

(i) By differentiation, find the range of values of xx for which the graph of y=xe−xy=xe^{-x} is decreasing. marks

(ii) Determine the range of values of xx for which the graph of y=xe−xy=xe^{-x} is decreasing and concave downwards. marks

(iii) The tangent to the curve at P(x,y)P(x,y) meets the yy-axis at R(0,h)R(0,h). Express hh in terms of xx, and find the greatest possible value of hh. marks

[2011/NYJC/Prelim/I/9]

Total marks: 12

Solution

Choose a part to attempt. Each page keeps the full question available and loads only that part’s working and exploration.

  1. (i) Decreasing functions →
  2. (ii) Concavity →
  3. (iii) intercept Tangent intercepts →
  4. (iii) greatest value Optimisation →