Q. 5 - Calculus

Question

Suppose that $f$ is defined on (−∞,∞) and differentiable everywhere except at $x=-2$ and $x=4$, and that $f\text{'}\left(x\right)=0$ only at $x=0$ and $x=5$. List all the critical points of $f$ and sketch a possible graph of $f$ . 

Step-by-Step Solution

Answer

$f\left(x\right)=0.2x^5-1.75x^4+0.67x^3+20x^2$

Step1. Given Information

$$\begin{gathered} \text{ The function is defined on }(-\mathrm{\infty },\mathrm{\infty })\text{ and is differentiable at all points except at }x=-2\text{ and }x=4\text{ and that } \\ {f}^{\mathrm{\prime }}\left(x\right)=0\text{ only at }x=0\text{ and }x=5. \end{gathered}$$

Step 2. Calculations

$$\begin{gathered} \text{ The critical points of the function, therefore will be } \\ x=-2;x=0;x=4;x=5 \\ \text{ Therefore, the differentiation of the function will be } \\ {f}^{\mathrm{\prime }}\left(x\right)=x(x+2)(x-4)(x-5) \\ \text{ The graph can be drawn as } \end{gathered}$$

$$\begin{gathered} \text{ Therefore, the function is } \\ f\left(x\right)=0.2x^5-1.75x^4+0.67x^3+20x^2 \end{gathered}$$