Differentiation of exponential and logarithmic functions — Question 36

Original QCE Vault practice · 5 marks

Q36 · Practice questionTechnology-freeComplex familiar5 marks

QUESTION 36 (5 marks)

Consider

f(x)=x^2e^{-x}

. Determine all stationary points for

x\ge0

and classify each as a local maximum or local minimum.

WORKED SOLUTION

5 marks

ANSWER

Local minimum at

(0,0)

; local maximum at

(2,4e^{-2})

Worked solution

f'(x)=e^{-x}(2x-x^2)=xe^{-x}(2-x)

. Thus

x=0,2

. For

x>0

, the derivative is positive on

(0,2)

and negative for

x>2

, so

x=2

is a local maximum. Extending to the endpoint,

f(x)\ge0

for

x\ge0

, so

(0,0)

is a local minimum on the stated domain.

Differentiates to

f'(x)=xe^{-x}(2-x)

[1 mark]

Finds stationary values

x=0,2

[1 mark]

Determines the coordinates.

[1 mark]

Classifies the point at

x=2

as a local maximum.

[1 mark]

Classifies the point at

x=0

as a local minimum on

x\ge0

[1 mark]

Practice question aligned to the current QCAA syllabus; review the worked solution and marking criteria.

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