Consider the function f(x)={x^5+0.4 x is less than or equal to 0 and -x+0.4 x>0. Use the value of f’(0) to guide you classify the point (0,0.4). Select all that apply.

Question

a) relative minimum b) absolute maximum c) absolute minimum d) relative maximum e) none of these

Answer 1

Define the functions

n(x) = x⁵ + 0.4

p(x) = -x + 0.4

Then

f(x) = n(x) for x <= 0, and f(x) = p(x) for x >0

Notice that

n(0) = p(0) = 0.4,

which makes f(x) continuous, although its derivative is not continuous.

First consider n(x).

n'(x) = 5x⁴

n'(0) = 0, but that does not make it maximum or minimum.

The important thing is that n'(x) >= 0, therefore

n(x) is non-decreasing, therefore

n(x) <= n(0) for all x <= 0, i.e,

n(x) has absolute maximum on the interval (-∞, 0] at x = 0.

Now consider p(x).

p'(x) = -1

p'(x) < 0, therefore

p(x) is always decreasing, therefore

p(x) <= p(0) for all x >= 0, therefore

p(x) has absolute maximum on the interval [0, ∞) at x = 0.

The conclusion is that the point (0, 0.4) is

absolute maximum for n(x) on (-∞, 0], and

absolute maximum for p(x) on [0, ∞).

Therefore it is absolute maximum for f.