Q. 43 - Calculus

Question

Use a sign chart for $f\text{'}\text{'}$ to determine the intervals on which each function f in Exercises 41–52 is concave up or concave down, and identify the locations of any inflection points. Then verify your algebraic answers with graphs from a calculator or graphing utility 

$f\left(x\right)=x^4-2x^3-5$

Step-by-Step Solution

Answer

The function has inflection points at $x=0,1$and is concave up on both $(-\infty ,0),(1,\infty )$ and concave down on $(0,1).$

Step 1. Given information.

The given function is $f\left(x\right)=x^4-2x^3-5$.

Step 2. Second Derivative.

On differentiating,

$$\begin{gathered} f^{\text{'}}\left(x\right)=\frac{d}{dx}\left(x^4-2x^3-5\right) \\ =4x^3-6x^2 \\ {f}^{\text{'}\text{'}}\left(x\right)=\frac{d}{dx}\left(4x^3-6x^2\right) \\ =12 x(x-1) \end{gathered}$$

Step 3. Sign chart.

Now,

$f\text{'}\text{'}\left(x\right)=0 at x=0,1.$

Therefore,

the chart will be,

$The function is cocave up in (-\infty ,0),(1,\infty )\text{ and concave down on }(0,1)$.

The inflexion points are $x=0,x=1.$

Step 4. Verification.

The graph of the function is,