Consider the function f(x)=12x^5+60x^4−100x^3+4. For this function there are four important intervals: (−INF,A], [A,B] ,[B,C] , and [C,INF) where A, B, and C are the critical numbers. Find A, B, and C.

At each critical number A, B, and C does f(x) have a local min, a local max, or neither? Type in your answer as LMIN, LMAX, or NEITHER.

Sorry to post another question but I have problems with this one too...

For B I have 0 and for C 1

I guess A will be local max B will be the local neither, c local min

I don't know which will be the value of A because when I graph it, it's actually a value really close to 0, and I have pluged, -1, -0.1, -0.2, -0.3 and -0.5 but none of them are the answer

A=-5

To find the critical numbers of a function, we need to determine when the derivative of the function equals zero or is undefined. In this case, we need to find the derivative of f(x) first.

Given f(x) = 12x^5 + 60x^4 - 100x^3 + 4, let's take the derivative.

f'(x) = d/dx (12x^5) + d/dx (60x^4) - d/dx (100x^3) + d/dx (4)

Differentiating each term using the power rule, we get:

f'(x) = 60x^4 + 240x^3 - 300x^2

Now, to find the critical numbers, we set the derivative equal to zero:

60x^4 + 240x^3 - 300x^2 = 0

Factoring out a common factor of 60x^2, we get:

60x^2(x^2 + 4x - 5) = 0

Setting each factor equal to zero, we have two cases:

Case 1: 60x^2 = 0
Solving for x, we get x = 0.

Case 2: x^2 + 4x - 5 = 0
Factoring or using the quadratic formula, we find two solutions: x = -5 or x = 1.

So, we have three critical numbers: A = -5, B = 0, and C = 1.

Now let's determine if each of these critical numbers corresponds to a local min, local max, or neither.

To do this, we can analyze the behavior of the derivative to the left and right of each critical number.

For A = -5:
- If we consider values of x less than -5, the derivative f'(x) is positive.
- If we consider values of x greater than -5, the derivative f'(x) is also positive.

Since the derivative is positive on both sides, f(x) does not have a local min or max at A.

For B = 0:
- If we consider values of x less than 0, the derivative f'(x) is negative.
- If we consider values of x greater than 0, the derivative f'(x) is positive.

Since the derivative changes sign at B, f(x) has a local min at B.

For C = 1:
- If we consider values of x less than 1, the derivative f'(x) is positive.
- If we consider values of x greater than 1, the derivative f'(x) is positive.

Since the derivative is positive on both sides, f(x) does not have a local min or max at C.

In conclusion, the critical numbers A, B, and C are -5, 0, and 1, respectively. At A, f(x) has neither a local min nor a local max. At B, f(x) has a local min. At C, f(x) has neither a local min nor a local max.