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Expert: - 9/17/2009


Question
I was given an assignment and would really appreciate if you can  help me with this question.

Find the critical points of function f(x) = x^3 [e^-6x]

Determine whether it is a maximum, minimum or inflexion point

Sketch the graph of y=f(x), indicating the critical points and the values where the graph cuts both the x-axis and y-axis.

thank you~

Answer

Graph
Questioner: Stella
Country: Singapore
Category: Calculus
Private: No
Subject: Finding the critical point
Question: I was given an assignment and would really appreciate if you can  help me with this question.

Find the critical points of function f(x) = x^3 [e^-6x]

Determine whether it is a maximum, minimum or inflexion point

Sketch the graph of y=f(x), indicating the critical points and the values where the graph cuts both the x-axis and y-axis.

thank you~
..............................................
Find the critical points of function f(x) = x^3 [e^-6x]

The key here is to remember that  e^(anything) is always positive, never zero.

You will find f'(x).  (product rule, anyone?)

f'(x) = ()() + ()()

f'(x) = (x^3)(-6e^-6x) + (3x^2)(e^-6x)

f'(x) = e^-6x[-6x^3 + 3x^2]

f'(x) = -3x^2e^-6x[2x - 1]

set that = 0, solve, you get  

x = 0  and  x = 1/2

Now you can do either:
A. The second derivative test:  Find  f''(x), evaluate it at x = 0, x = 1/2.

When  f'' is
-- Positive, you have a min.
-- Negative, you have a max.
-- Zero, you might not be sure.

B. The first derivative test.  Use the intervals:

(-inf,0)

(0,1/2)

(1/2, inf)

Pick a value of x in each and find f'(x) for it.  
The key here is to remember that  e^(anything) is always positive, never zero.

(-inf,0):  Pick  x = -1: f(-1) =  -3[-3 - 1] is positive. [RISING]

(0,1/2): Pick  x = 1/4:  f(1/4) = -3/4[1/2 - 1] is positive. [RISING]

(1/2, inf): Pick  x = 1: f(1) = -3(2 - 1) is negative. [FALLING]

I think x = 0 is an inflection point,  x = 1/2 is a max.

I will leave the other stuff to you:

cuts x-axis:  Solve  f(x) = 0.

cuts y-axis:  Find y = f(0)

Paul Klarreich

Expertise

All topics in first-year calculus including infinite series, max-min and related rate problems. Also trigonometry and complex numbers, theory of equations, exponential and logarithmic functions. I can also try (but not guarantee) to answer questions on Analysis -- sequences, limits, continuity.

Experience

I taught all mathematics subjects from elementary algebra to differential equations at a two-year college in New York City for 25 years.

Education/Credentials
(See above.)

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