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Sketch the graph of a function that satisfies the given conditions given as follows:

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

Figure 1

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.

Figure 2

Images/Mathematical drawings are created with Geogebra.

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