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

If this part uses an earlier result, revisit (i) Decreasing functions or open all working together.

(ii)

Concavity

Determine the range of values of xx for which the graph is decreasing and concave downwards.

Explore this idea

Hint

  1. Concave downwards means d2ydx2<0\frac{d^2y}{dx^2}<0.

  2. Combine with part (i).

Answer · (ii)

1<x<21<x<2.

Explore the idea

Open interactive exploration

Falling and bending down: 1 < x < 2

Switch on the maximum and the point of inflexion.

Switch on a feature to add it to the sketch and see how it is found. A full-mark sketch shows all of them.

  • Y = xe⁻ˣ
  • X = 1 and x = 2
Ideas to try
Between the two lines, is the curve going down and bending down?