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 (ii) Concavity or open all working together.

(iii) intercept

Tangent intercepts

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.

Explore this idea

Hint

  1. The tangent has gradient e−x(1−x)e^{-x}(1-x) and passes through P(x,xe−x)P(x,xe^{-x}).

  2. Put X=0X=0 in the tangent equation.

Answer · (iii) intercept

h=x2e−xh=x^2e^{-x}.

Explore the idea

Open interactive exploration

Where the tangent meets the y-axis

Drag the point P. The amber tangent hits the y-axis at height h = x²e⁻ˣ.

Position along the curve(2, 0.27067)x = 2
Try an example
Tangent gradient
-0.13534

At x=2x=2: the tangent meets the yy-axis at h=4e−2≈0.54h=4e^{-2}\approx0.54.

For finite nonzero gradients, tangent gradient × normal gradient = −1. Both axes use equal scales.

  • Y = xe⁻ˣ
  • Tangent at P
Key idea

Each position of PP gives one value of hh: the function h(x)=x2e−xh(x)=x^2e^{-x}.

Ideas to try
Slide to the left. Does the intercept keep growing?