Decreasing curves and the largest tangent intercept

y=xe^{-x}

Total marks: 12

Solution

(i)

Decreasing functions

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

Explore this idea

Hint

Differentiate with the product rule.
Decreasing means $\frac{dy}{dx}<0$ .

Answer · (i)

x>1

Explore the idea

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: $x>1$ .

Ideas to try

Find the point where the gradient is exactly 0.

(ii)

Concavity

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

Explore this idea

Hint

Concave downwards means $\frac{d^2y}{dx^2}<0$ .
Combine with part (i).

Answer · (ii)

1<x<2

Explore the idea

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?

(iii) intercept

Tangent intercepts

The tangent to the curve at $P(x,y)$ meets the $y$ -axis at $R(0,h)$ . Express $h$ in terms of $x$ .

Explore this idea

Hint

The tangent has gradient $e^{-x}(1-x)$ and passes through $P(x,xe^{-x})$ .
Put $X=0$ in the tangent equation.

Answer · (iii) intercept

h=x^2e^{-x}

Explore the idea

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 $x=2$ : the tangent meets the $y$ -axis at $h=4e^{-2}\approx0.54$ .

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 $P$ gives one value of $h$ : the function $h(x)=x^2e^{-x}$ .

Ideas to try

Slide to the left. Does the intercept keep growing?

(iii) greatest value

Optimisation

Find the greatest possible value of $h$ .

Explore this idea

Hint

Differentiate $h=x^2e^{-x}$ and set it to 0.
Classify the stationary points and check the ends.

Answer · (iii) greatest value

No greatest value on the printed unrestricted domain. h\to\infty as x\to-\infty. If x\geq0 (or x>1) is intended, the maximum is 4e⁻² at x=2.

Explore the idea

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

4e^{-2}

the greatest value if

x

can be negative?

Decreasing curves and the largest tangent intercept

Solution

Decreasing functions

Product rule

Sign

Solve

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

Concavity

Second derivative

Concave down

Combine with $x>1$

Falling and bending down: 1 < x < 2

Tangent intercepts

Tangent at $P$

Intercept at $X=0$

Simplify

Where the tangent meets the y-axis

Optimisation

Differentiate

Stationary points

Nature

The domain

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

Product rule

Sign

Solve

Second derivative

Concave down

Combine with x>1x>1x>1

Tangent at PPP

Intercept at X=0X=0X=0

Simplify

Differentiate

Stationary points

Nature

The domain

Combine with $x>1$

Tangent at $P$

Intercept at $X=0$