Complete the sentence with one of the choices listed below. The function f(x) = 4e^-2x has_______?

Question

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exactly two inflection points, one local maximum and no local minimum


exactly three inflection points, one local maximum and one local minimum


exactly two inflection points, no local maximum and one local minimum


no inflection points, no local maximum and no local minimum


properties not described above

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Answer 1

no inflection points, no local maximum and no local minimum

This is the only answer that does not require a maximum or a minimum to exist. All equations of the form ae^bx are monotonic. if b is positive, they are monotonically increasing. If b is negative they are monotonically decreasing. As a result no maxima or minima can exist.

An inflection point exists when the second derivative of a function has a value of zero.

The first derivative is 4(-2)e^-2x
The second derivative is 4(-2)(-2)e^-2x = 16e^-2x. This an exponential function and is therefore never equal to zero.

Alternatively, you can solve for a value of zero and get the result
x = -ln(0)/2 - which is an undefined expression
(there is no natural log of zero).

Answer 2

f(x) = 4e^(-2x)

f'(x) = -8e(-2x)

f"(x) = 16e(-2x)

Since e^(-2x) is greater than zero, there are no critical points or inflection points. The function is decreasing and concave up.

So the fourth option is the correct one, there are no inflection points, no local maximum and no local minimum.

Answer 3

properties not described above