Help with Critical Number

[letsgetaway]

I'm having trouble classifying the critical number of the following function. I took the derivative of the function but I wasn't able to factor the derivative to find where it equals zero or does not exist to use for a sign chart to show me the min/max.

The derivative is f'(x) = x^4 - x^3 - 7x^2 + 13x - 6

I'm not sure how to proceed from here.

The value x = 1 is a critical number for the function

Classify this critical number.
a) local minimum
b) neither
c) local maximum

 

[morson]

Forget the factor theorem?

Let P(x) = x^4 - x^3 - 7x^2 + 13x - 6 = 0

P(1) = 0 ==> (x - 1) is a factor of P(x)

P(x) = x^4 - x^3 - 7x^2 + 13x - 6 = (x^3 - 7x + 6)(x - 1), by division

Let D(x) = x^3 - 7x + 6 = 0

D(1) = 0 ==> (x - 1) is a factor of D(x)

x^3 - 7x + 6 = (x^2 + x - 6)(x - 1)

==> P(x) = (x^2 + x - 6)(x - 1)(x - 1)

You should be able to complete the factorising (it's simple).

As for classifying the stationary points, use the second/first derivative tests.

 

[letsgetaway]

Thanks. I was able to figure it out using your help.