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

(i)

Decreasing functions

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

Explore this idea

Hint

  1. Differentiate with the product rule.

  2. Decreasing means dydx<0\frac{dy}{dx}<0.

Answer · (i)

x>1x>1.

Explore the idea

Open interactive exploration

y = xe⁻ˣ rises, peaks at x = 1, then falls

The shaded stretch is where the gradient is positive.

Position along the curvex = 2
Try an example
Test a critical position

Buttons use full-precision reviewed positions; displayed coordinates are rounded.

f(x)
0.27067
Gradient f′(x)
-0.13534
negative

For x>1 the gradient is negative: the curve falls.

Selected values are within the original drawing windows.

  • Y = xe⁻ˣ
  • Tangent
  • Dy/dx, and where the curve rises
  • Join P to Q; the dashed legs show Δx and Δy
Key idea

The curve is decreasing exactly where the gradient graph is below the axis: x>1x>1.

Ideas to try
Find the point where the gradient is exactly 0.