Final answer:
To find where f(x)=-6 x³+27 x²+720 x-3 is increasing, calculate its derivative f'(x), solve the inequality f'(x) > 0, and then determine the interval of x that satisfies this condition.
Step-by-step explanation:
To determine the interval where the function f(x)=-6 x³+27 x²+720 x-3 is increasing, we need to find the derivative of the function, f'(x), and identify where the derivative is positive. The derivative represents the slope of the function, so if it's positive, the function is increasing.
Step by Step Solution:
Calculate the derivative of f(x): f'(x) = d/dx(-6 x³+27 x²+720 x-3) which simplifies to -18 x²+54 x+720.
Find the values of x where f'(x) is positive. This is equivalent to solving the inequality -18 x²+54 x+720 >0.
Use any method such as factoring, completing the square, or using the quadratic formula to solve this inequality.
Choose x values within the interval that satisfy the inequality and test them in f'(x) to confirm that they produce a positive result.
The results of these steps give us the interval on which f(x) is increasing.