**Here are the conditions for this problem:**\[ f'(0) = f'(2) = f'(4) = 0 \] \[ f'(x) \gt 0\ if \left\{ \begin {array} x \lt 0 \\ 2 \lt x \lt 4 \end {array} \right. \] \[ f'(x) \lt 0\ if \left\{ \begin {array} 0 \lt x \lt 2 \\ x \gt 4 \end {array} \right. \] \[ f”(x) \gt 0\ if\ 1 \lt x \lt 3 \] \[ f”(x) \lt 0\ if \left\{ \begin {array} x \lt 1 \\ x \gt 3 \end {array} \right. \] The question aims to find the

**graph**of a

**function**that satisfies the

**given conditions.**The concepts needed for this question are

**derivative, maxima, minima,**and

**2nd derivative test.**A

**local maximum**is the

**highest point**on the graph of the function where the

**first derivative**is

**zero,**and the function starts

**decreasing**after that point. A

**local minimum**is the

**lowest point**on the function’s graph where the

**first derivative**is

**zero,**and the function starts to

**increase**after that point. The

**second derivative**test is performed on any given function to check for

**local extremas.**The

**2nd derivative test**checks whether there are

**local maxima**or

**local minima**at a certain

**point**of the given function. Let

**c**is the given point on the graph of the given

**function f**, and we want to check if it contains

**local maxima**or

**minima.**First, we take the

**first derivative**of the

**function f at point c.**\[ f'(c) = 0 \] When the

**function’s first derivative**is

**zero**at

**point**

**c**, this means that the function has a

**critical point**at

**c**. Then we take the

**2nd derivative**and check its value at

**c**, and the following situations can occur: \[ f'(c) = 0, \hspace{0.2in} f”(c) \lt 0 \hspace{0.2in} Local\ Maximum \] \[ f'(c) = 0, \hspace{0.2in} f”(c) \gt 0 \hspace{0.2in} Local\ Minimum \] \[ f'(c) = 0, \hspace{0.2in} f”(c) = 0 \hspace{0.2in} Inconclusive \]

## Expert Answer

The given condition represents that there are**critical points**at

**x=0,**

**2, and 4**. We can assume that there will be either

**local minima**or

**local maxima**existing at these points. We can also see from the given conditions that the

**function increases**when

**x**is

**less than zero**and when

**x**is

**greater than**

**2**. But the function

**decreases**when

**x**is

**greater**than

**zero**and when

**x**is

**greater**than

**4**. Using the above information from the conditions, we can conclude that there will be local maxima at x=0 as the function increases before and decreases after this point. We can also conclude that there will be a

**local minima**at

**x=2**as the function

**decreases**at this point and

**increases**after it. Similarly, there will be a

**local maxima**at

**x=4.**We can also conclude that there are

**points**of

**inflections**at

**x=1**and

**x=3.**Using this information, we can

**draw**an approximation of the

**function graph**shown in Figure 1.

## Numerical Result

## Example

Sketch the**graph**with the following conditions: \[ f'(1) = 0 \] \[ f'(x) \gt 0\ if\ x \lt 1 \] \[ f'(x) \lt 0\ if\ x \gt 1 \] We can conclude that there is a

**critical point**at

**x=1.**The function

**f increases**before

**x=1,**and it

**decreases**after

**x=1.**We can draw an

**approximation**of the

**graph**of the

**function**shown in Figure 2 below.

*Images/Mathematical drawings are created with Geogebra.*

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