Final answer:
To find the critical points for the function f(x) = (3/4)x^4 + 6x^3 - (75/2), the derivative f'(x) is set to zero resulting in critical points at x=0 and x=-6. The First Derivative Test is applied to determine the nature of these points. Sign analysis of f'(x) reveals the intervals of increase and decrease.
Step-by-step explanation:
To find the critical points of a function, we need to take the derivative of the function and set it equal to zero. For the function f(x) = \( \frac{3}{4} x^4 + 6x^3 - \frac{75}{2} \), let's find the first derivative, f'(x), which is given by:
f'(x) = 3x^3 + 18x^2.
We set the derivative equal to zero to find the critical points:
0 = 3x^3 + 18x^2.
Solving for x, we get:
x^2(3x + 18) = 0,
which implies x = 0 or x = -6.
These are our critical points. To determine if these points correspond to a maximum, minimum, or inflection point, we can apply the First Derivative Test. This involves analyzing the sign of f'(x) to the left and right of each critical point:
- If f'(x) changes from positive to negative, the critical point is a local maximum.
- If f'(x) changes from negative to positive, the critical point is a local minimum.
- If f'(x) does not change sign, the critical point is neither (inflection point).
To find intervals where the function is increasing or decreasing, we look at the sign of f'(x):
- If f'(x) > 0, the function is increasing.
- If f'(x) < 0, the function is decreasing.
By performing the sign analysis for each interval divided by the critical points, we'll be able to determine where the function is increasing or decreasing.