How many critical values does the function

f(x) = 3x^4 + 4x^3

have?

I know f' has to = zero.

Is the answer 2?

(x=0,−1)

3x^4 + 4x^3 =0

x^3(3x+4) = 0
x = 0 or x = -4/3

looks like two x-intercepts

f ' (x) = 12x^3 + 12x^2
= 0 for a max/min of f(x)
12x^3 + 12x^2 = 0
x^2(12x + 12) = 0
x = 0 , f(0) = 0
x = -1, f(-1) = 3 + 4(-1) = -1
looks like (-1, -1) is a minimum , and we will consider (0,0) later

f ''(x) = 36x^2 + 24x
= 0 at a point of inflection
x(36x + 24) = 0
x = 0 or x = -24/36 = -2/3
so (0,0) and (-2/3, ....) are points of inflection

confirmation:
http://www.wolframalpha.com/input/?i=plot+y+%3D+3x%5E4+%2B+4x%5E3

To determine the critical values of a function, we need to find the values of x where the derivative of the function is equal to zero or undefined. The derivative of a function gives us information about the slope or rate of change of the function.

In this case, we have the function:
f(x) = 3x^4 + 4x^3

To find the derivative, we use the power rule:
f'(x) = 12x^3 + 12x^2

Now, we need to find the values of x where the derivative is equal to zero or undefined. Setting the derivative equal to zero gives us:
12x^3 + 12x^2 = 0

Factoring out 12x^2, we have:
12x^2(x + 1) = 0

Setting each factor equal to zero, we get:
12x^2 = 0 ==> x = 0
x + 1 = 0 ==> x = -1

So, the values x = 0 and x = -1 are the critical values of the function f(x) = 3x^4 + 4x^3.

Therefore, the function has two critical values.