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 (iii) intercept Tangent intercepts or open all working together.

(iii) greatest value

Optimisation

Find the greatest possible value of hh.

Explore this idea

Hint

  1. Differentiate h=x2e−xh=x^2e^{-x} and set it to 0.

  2. Classify the stationary points and check the ends.

Answer · (iii) greatest value

No greatest value on the printed unrestricted domain.

h→∞ as x→−∞. If x≥0 (or x>1) is intended, the maximum is 4e⁻² at x=2.

Explore the idea

Open interactive exploration

h(x) = x²e⁻ˣ has a local maximum but no greatest value

Switch on the turning points and look at the left tail.

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

  • H(x) = x²e⁻ˣ
  • Local maximum value
Ideas to try
Is 4e−24e^{-2} the greatest value if xx can be negative?