Q .28 - Calculus

Question

Use a sign chart for ${\mathit{f}}^{\mathbf{\text{'}}}$ to determine the intervals on which each function $\mathit{f}$ is increasing or decreasing. Then verify your algebraic answers with graphs from a calculator or graphing utility.

$\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}{\mathit{x}}^{\mathbf{3}}\mathbf{+}\mathbf{4}{\mathit{x}}^{\mathbf{2}}\mathbf{+}\mathbf{4}\mathit{x}\mathbf{-}\mathbf{5}$

Step-by-Step Solution

Answer

Ans : 

Intervals of the given function is $\mathbf{(}\mathbf{-}\mathbf{\infty }\mathbf{,}\mathbf{-}\mathbf{2}\mathbf{]}\mathbf{\cup }\mathbf{[}\frac{\mathbf{-}\mathbf{2}}{\mathbf{3}}\mathbf{,}\mathbf{\infty }\mathbf{)}$

Increasing at $\mathbf{(}\mathbf{-}\mathbf{\infty }\mathbf{,}\mathbf{-}\mathbf{2}\mathbf{]}\mathbf{\cup }\mathbf{[}\frac{\mathbf{-}\mathbf{2}}{\mathbf{3}}\mathbf{,}\mathbf{\infty }\mathbf{)}$$\mathbf{(}\mathbf{-}\mathbf{\infty }\mathbf{,}\mathbf{-}\mathbf{2}\mathbf{]}\mathbf{\cup }\mathbf{[}\frac{\mathbf{-}\mathbf{2}}{\mathbf{3}}\mathbf{,}\mathbf{\infty }\mathbf{)}$

Decreasing at $\mathbf{(}\mathbf{-}\mathbf{2}\mathbf{,}\frac{\mathbf{-}\mathbf{2}}{\mathbf{3}}\mathbf{)}$

Step 1. Given information:

$\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}{\mathit{x}}^{\mathbf{3}}\mathbf{+}\mathbf{4}{\mathit{x}}^{\mathbf{2}}\mathbf{+}\mathbf{4}\mathit{x}\mathbf{-}\mathbf{5}$

Step 2. Finding the derivative of the function:

$$\begin{gathered} f\left(x\right)=x^3+4x^2+4x-5 \\ {f}^{\text{'}}\left(x\right)=3x^2+8x+4 \\ let, f^{\text{'}}\left(x\right)=0 \\ \therefore 3x^2+8x+4=0 \\ 3x^2+6x+2x+4=0 \\ 3x(x+2)+2(x+2)=0 \\ \boxed{\overline{)3x+2=0 \\ x=\frac{-2}{3}}} \\ \boxed{x+2=0 \\ x=-2} \\ x=\frac{-2}{3},-2 \end{gathered}$$

Step 3. Inserting the root points on the number line(Sign chart):

After inserting the root values we can find the increasing and decreasing intervals of the given function. 

Intervals of the given function is $\mathbf{(}\mathbf{-}\mathbf{\infty }\mathbf{,}\mathbf{-}\mathbf{2}\mathbf{]}\mathbf{\cup }\mathbf{[}\frac{\mathbf{-}\mathbf{2}}{\mathbf{3}}\mathbf{,}\mathbf{\infty }\mathbf{)}$

Increasing at $\mathbf{(}\mathbf{-}\mathbf{\infty }\mathbf{,}\mathbf{-}\mathbf{2}\mathbf{]}\mathbf{\cup }\mathbf{[}\frac{\mathbf{-}\mathbf{2}}{\mathbf{3}}\mathbf{,}\mathbf{\infty }\mathbf{)}$

Decreasing at$\mathbf{(}\mathbf{-}\mathbf{2}\mathbf{,}\frac{\mathbf{-}\mathbf{2}}{\mathbf{3}}\mathbf{)}$

Step 4. Verifying algebraic answers with graphs :