Q. 27 - 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.

$\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\mathbf{2}{\mathit{x}}^{\mathbf{3}}\mathbf{-}\mathbf{9}{\mathit{x}}^{\mathbf{2}}\mathbf{+}\mathbf{1}$

Step-by-Step Solution

Answer

Ans: 

Intervals of the given function is $\mathbf{(}\mathbf{-}\mathbf{\infty }\mathbf{,}\mathbf{0}\mathbf{]}\mathbf{\cup }\mathbf{[}\mathbf{3}\mathbf{,}\mathbf{\infty }\mathbf{)}$

Increasing at $\mathbf{(}\mathbf{-}\mathbf{\infty }\mathbf{,}\mathbf{0}\mathbf{]}\mathbf{\cup }\mathbf{[}\mathbf{3}\mathbf{,}\mathbf{\infty }\mathbf{)}$

Decreasing at  $\mathbf{(}\mathbf{0}\mathbf{,}\mathbf{3}\mathbf{)}$

Step 1. Given information:

$\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\mathbf{2}{\mathit{x}}^{\mathbf{3}}\mathbf{-}\mathbf{9}{\mathit{x}}^{\mathbf{2}}\mathbf{+}\mathbf{1}$

Step 2. Finding the derivative of the function:

$$\begin{gathered} f\left(x\right)=2x^3-9x^2+1 \\ {f}^{\text{'}}\left(x\right)=6x^2-18x \\ let, f^{\text{'}}\left(x\right)=0 \\ \therefore 6x^2-18x=0 \\ 6x(x-3)=0 \\ x=0,3 \end{gathered}$$

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 $\mathbf{(}\mathbf{-}\mathbf{\infty }\mathbf{,}\mathbf{0}\mathbf{]}\mathbf{\cup }\mathbf{[}\mathbf{3}\mathbf{,}\mathbf{\infty }\mathbf{)}$

Increasing at $\mathbf{(}\mathbf{-}\mathbf{\infty }\mathbf{,}\mathbf{0}\mathbf{]}\mathbf{\cup }\mathbf{[}\mathbf{3}\mathbf{,}\mathbf{\infty }\mathbf{)}$

Decreasing at $\mathbf{(}\mathbf{0}\mathbf{,}\mathbf{3}\mathbf{)}$

Step 4. Verifying algebraic answers with graphs :