Question

4. Which could be the function of the following graph?

A. f(x) = (x + 2)(x-2)³(x-5)²
B. f(x) = (x + 2)³(x - 2)²(x - 5)²(x-4)
C. f(x) = (x+2)³(x - 2)²(x - 5)²
D. f(x) = (x + 2)²(x-2)(x - 5)³​

Answer 1

none of the above

best choice: D

Explanation:

You want a possible function for the given graph.

Zeros

The graph shows a function with zeros at -2, +2, and +5. The multiplicity of the zero at -2 is even, which is why the x-axis is not crossed there.

The graph shows a function with odd multiplicity of the zeros at +2 and +5. However, an odd exponent greater than 1 will produce a "flat spot" at the zero crossing. There isn't one at either place, so we must conclude ...

sadly, None of the Above

Best choice

The only function choice with a factor of (x+2) having even multiplicity is ...

D. f(x) = (x + 2)²(x-2)(x - 5)³

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Additional comment

The attached graph shows an approximation of the given graph by ...

f(x) = (x +2)²(x -2)(x -5)/10.5

The vertical scaling is necessary to get the peaks near where they are shown.

The second attachment shows a scaled version of function D. Its relative peak values are all wrong, along with the zero crossing at x=5 and the vertical scaling.