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Analyzing Exponential Functions

Now we’ll take a look at an example where we analyze an exponential function. The method is as follows:

Rule

Analyzing Exponential Functions

1.
Find the zeros.
2.
Find the stationary points.
3.
Find the inflection points.

Example 1

Analyze the function f(x) = 2x2 ex

Example of analysis of exponential function

Example of analysis of exponential function

1.
Find the zeros by setting f(x) = 0:
2x2 ex = 0

The zero product property gives that 2x2 = 0 or ex = 0. However, ex is always positive, so 2x2 = 0 x2 = 0 x = 0

This gives a zero at the origin (0, 0).

2.
Find the maxima and minima by setting f(x) = 0.

First, find the derivative of f(x) = 2x2 ex: f(x) = 4x ex + 2x2 ex = ex (4x + 2x2) = 2x ex(2 + x)

Then, find where f(x) is equal to 0:

2x ex(2 + x) = 0

Again, ex is always positive, so 2x = 0 x = 0 2 + x = 0 x = 2

You then need the corresponding y-values to find the point. You do this by inputting your x-values back into the main function f(x):

y = f(0) = 2 02 e0 = 0 y = f(2) = 2(2)2 e2 = 8e2 = 8 e2

y = f(0) = 2 02 e0 = 0 y = f(2) = 2(2)2 e2 = 8e2 = 8 e2

You now need to determine which point is a maximum and which is a minimum. You do that by drawing a sign chart.

Sign chart of exponential function

Sign chart of exponential function

From this, you see that the maximum is (2, 8 e2 ) and the minimum is (0, 0).
3.
Find the inflection points by setting f(x) = 0.

First, you find the second derivative by differentiating f(x) = ex(4x + 2x2):

f(x) = ex (4x + 2x2) + ex(4 + 4x) = ex (4x + 2x2 + 4 + 4x) = ex (2x2 + 8x + 4)

f(x) = ex (4x + 2x2) + ex(4 + 4x) = ex (4x + 2x2 + 4 + 4x) = ex (2x2 + 8x + 4)

Then, let f(x) = 0 and solve the equation:
ex (2x2 + 8x + 4) = 0

As ex is always positive, you get

2x2 + 8x + 4 = 0

You solve this using the quadratic formula and get the solutions x 0.6 and x 3.4. You find the corresponding y-values by putting your new x-values back into the main function f(x). You then get: y = f(3.4) = 2 (3.4)2 e3.4 0.772 y = f(0.6) = 2(0.6)2 e0.6 = 8e0.6 0.395

which means that you have inflection points at (3.4,0.772) and (0.6,0.395).

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How to Analyze the Cosine Function