Q. 1 TB - Calculus

Question

Finding sign intervals: Consider the function$f\left(x\right)=x^3-4x^2-12x$. Find the intervals on which f is positive and the intervals on which f is negative.

Step-by-Step Solution

Answer

Ans: 

Step 1. GIven information:

$f\left(x\right)=x^3-4x^2-12x$

Step 2. Finding the roots of the function:

$$\begin{gathered} f\left(x\right)=x^3-4x^2-12x \\ let, f\left(x\right)=0 \\ {x}^3-4x^2-12x=0 \\ {x}^3-6x^2+2x^2-12x=0 \\ {x}^2\left(x-6\right)+2x(x-6)=0 \\ \left(x-6\right)(x^2+2x)=0 \\ \left(x-6\right)\left(x\right(x+2\left)\right)=0 \\ x(x-6)(x+2)=0 \\ so, roots are 0,-2,6 \end{gathered}$$

Step 3. Finding the positive and negative interval of the function:

a