Question

Consider the equation below. f(x) = 4x3 + 6x2 − 24x + 3 (a) find the intervals on which f is increasing. (enter your answer using interval notation.)

Answer 1

The function f(x) = 4x^3 + 6x^2 - 24x + 3 is increasing on the interval (-2/3, 1.5), where the derivative of the function is positive.

Step-by-step explanation:

To find the intervals on which the function f(x) = 4x^3 + 6x^2 − 24x + 3 is increasing, we first need to calculate its derivative, which will give us the slope of the tangent line to the function at any point x. The function is increasing wherever its derivative is positive.

The derivative of f with respect to x is:
f'(x) = 12x^2 + 12x - 24.

To find where the function is increasing, set the derivative greater than zero:

12x^2 + 12x - 24 > 0

Solve this inequality to find the intervals for x. Factoring the quadratic, we get:

(2x - 3)(6x + 4) > 0

The zeros of this equation are x = 1.5 and x = -2/3. These values divide the real number line into intervals. Test a value from each interval to see where the function is increasing:

• If x < -2/3, for example x = -1, the derivative becomes negative, indicating the function is decreasing.
• If -2/3 < x < 1.5, for example x = 0, the derivative is positive, indicating the function is increasing.
• If x > 1.5, for example x = 2, the derivative is again negative, indicating the function is decreasing.

Thus, the function f(x) is increasing in the interval (-2/3, 1.5).

Answer 2

with increasing f from the equation stated above, we get

For increasing

F(x) > 0 => 12 x^2 + 12 x – 24 > 0

ð X^2 + x -1 > 0

ð (x + 2)(x – 1 ) > 0

ð X € (negative infinity, -2) union (1, infinity)