find the interval at which the function increases and decreases

x^3\3 +x^2\2 +x +100

To determine the intervals in which a function increases or decreases, we need to find the critical points first. Critical points are points where the derivative of the function is either zero or undefined. So, let's find the derivative of the function first.

Given function: f(x) = x^3/3 + x^2/2 + x + 100

To find the derivative, we can simply differentiate each term of the function:

f'(x) = (1/3) * d/dx(x^3) + (1/2) * d/dx(x^2) + d/dx(x) + d/dx(100)

Differentiating each term:
f'(x) = (1/3) * 3x^2 + (1/2) * 2x + 1 + 0

Simplifying the derivative:
f'(x) = x^2 + x + 1

Now that we have the derivative, let's solve for the critical points by setting f'(x) equal to zero:

x^2 + x + 1 = 0

Unfortunately, this quadratic equation does not have any real solutions. So, there are no critical points in this case.

Since there are no critical points, the function f(x) = x^3/3 + x^2/2 + x + 100 either increases or decreases throughout its entire domain.