let f'(x)=4x^3-12x^2
a. find three different functions with derivative equal to f'(x)
b. find the behavior
f(x) = x^4 + 4x^3 + c , where c is any constant. Put in any c value you want.
They would differ from each other by a vertical shift.
a. To find three different functions with a derivative equal to f'(x), we need to integrate f'(x). Let's integrate:
∫f'(x) dx = ∫(4x^3 - 12x^2) dx
To integrate this, we use the power rule of integration, which states that for a term ax^n, the integral is (a/(n+1)) * x^(n+1). Applying this rule to each term, we get:
∫4x^3 dx = (4/(3+1)) * x^(3+1) = x^4
∫-12x^2 dx = (-12/(2+1)) * x^(2+1) = -4x^3
Therefore, the integral of f'(x) is:
∫f'(x) dx = x^4 - 4x^3 + C
where C is the constant of integration.
By choosing different values for C, we can obtain different functions with the same derivative. For example:
Function 1: f(x) = x^4 - 4x^3 + 10
Function 2: f(x) = x^4 - 4x^3 - 5
Function 3: f(x) = x^4 - 4x^3 + 2
All these functions have a derivative equal to f'(x) = 4x^3 - 12x^2.
b. To determine the behavior of the given function, we analyze its derivative:
f'(x) = 4x^3 - 12x^2
The derivative tells us about the slope of the function at various points. To identify the behavior, we look at the sign of f'(x) in different regions.
To do this, we solve the inequality f'(x) > 0 to find where the function is increasing, and f'(x) < 0 to find where it is decreasing.
Let's solve f'(x) > 0:
4x^3 - 12x^2 > 0
Factor out common terms:
4x^2 (x - 3) > 0
Solve each factor separately:
4x^2 > 0 => x^2 > 0 => x ≠ 0
x - 3 > 0 => x > 3
Thus, the function is increasing when x > 3 and x ≠ 0.
Now let's solve f'(x) < 0:
4x^3 - 12x^2 < 0
Factor out common terms:
4x^2 (x - 3) < 0
Solve each factor separately:
4x^2 < 0 => x^2 < 0
x - 3 < 0 => x < 3
Since x^2 cannot be negative, the inequality x^2 < 0 has no real solutions. Therefore, the function is not decreasing for any value of x.
In summary, the behavior of the function f(x) = x^4 - 4x^3 + C (where C is any constant) is as follows:
- The function is increasing for x > 3 and x ≠ 0.
- The function does not have any regions of decreasing behavior.