How do you find the critical numbers for #f(x) = 3x^4 + 4x^3 - 12x^2 + 5# to determine the maximum and minimum?

Question

Scarlett Allison · Accepted Answer

Critical points occur at #x in {-2,0,1}#

Critical point occur where the derivative of the function is equal to zero.

Given
#color(white)("XXX")f(x)=3x^4+4x^3-12x^2+5#

First derivative:
#color(white)("XXX")f'(x)=12x^3+12x^2-24x#
which can be factored as
#color(white)("XXX")=(12)(x)(x^2+1-2)#

#color(white)("XXX")=(12)(x)(x+2)(x-1)#

Which implies the critical points (when #f'(x)=0#) occur when
#color(white)("XXX")x=0#
#color(white)("XXX")(x+2)=0color(white)("xxx")rarr x=-2# and
#color(white)("XXX")(x-1)=0color(white)("xxx")rarr x=+1#

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

While not explicitly asked for in this question,
you can determine if each of these critical points is a minimum or maximum by evaluating the second derivative at each critical value.
Results greater than zero indicate a local minimum;
results less than zero indicate a local maximum;
results equal to zero indicate an inflection point.

#{: (f''(x),=36x+24x-24,,), ("at "x=-2,= +72,>0,"minimum"),("at "x=0,=-24, <0, "maximum"),("at "x=+1,=+36,>0,"maximum") :}#

Here is the graph for verification:

Luke Alvarez · Answer

undefined To find the critical numbers of $ f(x) = 3x^4 + 4x^3 - 12x^2 + 5 $, we first find its derivative, $ f'(x) $. Then, we set $ f'(x) $ equal to zero and solve for $ x $. These solutions are the critical numbers. Finally, we use the first or second derivative test to determine if these critical numbers correspond to maximum or minimum points.

$$ f'(x) = 12x^3 + 12x^2 - 24x $$

Setting $ f'(x) = 0 $, we have:

$$ 12x^3 + 12x^2 - 24x = 0 $$

Factoring out $ 12x $, we get:

$$ 12x(x^2 + x - 2) = 0 $$

Now, solving $ x^2 + x - 2 = 0 $ for $ x $, we find:

$$ x^2 + x - 2 = 0 $$
$$ (x + 2)(x - 1) = 0 $$

So, the critical numbers are $ x = -2 $, $ x = 0 $, and $ x = 1 $.

We can now use the first or second derivative test to determine if these correspond to maximum or minimum points.