f(x)=(2^x)/x^2

Question

a.) Find f'(x)

b.) Find the intervals where f is increasing and the intervals f is decreasing.

c.) Use the fact e=2.71 to find two integers that ln(2) is between. Is 2/ln(2) greater than or less than 2.

d.) find f''(x)

e.) Find intervals where f is concave up and intervals where f is concave down. 

Answer 1

(Answers in bold for convenience)

For part (a), I prefer to rewrite f(x) as a product and calculate the derivative using the product rule:

f(x) = 2^x • x^-2

And so

f’(x) = 2^x • ln(2) • x^-2 – 2 • 2^x • x^-3

For part (b), factor f’(x) to calculate critical points of the function:

f’(x) = 2^x • x^-3 • (x ln(2) – 2)

The derivative fails to exist at x = 0, and equals 0 at x = 2 / ln(2). By using the first derivative test, we conclude the following intervals of increase and decrease:

f(x) increases over (−∞, 0) and over [2 / ln(2), +∞)

f(x) decreases over (0, 2 / ln(2)]

Part (c) is made simpler with the observation that 1 < 2 < e. Since ln(x) is a one-to-one function, the order is preserved when performing the natural log:

ln(1) < ln(2) < ln(e)

But ln(1) = 0 and ln(e) = 1, which means ln(2) is between 0 and 1.

In particular, this means that 1 / ln(2) > 1, and therefore 2 / ln(2) > 2.

For part (d) we differentiate the factored form of f’(x) we calculated in part (b) and obtain

f”(x) = (2^x • ln(2) • x^-3 – 3 • 2^x • x^-4) • (x ln(2) – 2) + 2^x • x^-3 • ln(2)

For part (e) we factor f”(x) and identify its critical points:

f”(x) = 2^x • x^-4 • ((x ln(2) – 3) • (x ln(2) – 2) + x ln(2))

The second derivative fails to exist at x = 0, but no other critical values are obvious yet. We expand the last expression in the parentheses as follows:

(x ln(2) – 3) • (x ln(2) – 2) + x ln(2) = (x ln(2))² – 4x ln(2) + 6

This is a quadratic expression in x ln(2) expressible in the form

(x ln(2) – 2)² + 2

which is always positive. This means that f(x) has no inflection points, and therefore f(x) is concave up over its entire domain:

f(x) is concave up over (–∞, 0) and over (0, +∞)

f(x) is never concave down