Question

What is the sign of f on the interval 0 < x < 2

f(x) = - x(x+5)(x+1)(x-2)

Always Positive
Always Negative
Sometimes Positive and Sometimes Negative
Always Greater than 0
Always less than -1

Answer 1

For the function f(x) = -x(x+5)(x+1)(x-2) on the interval 0 < x < 2, the sign is always negative.

Consider the function f(x) = -x(x+5)(x+1)(x-2) on the interval 0 < x < 2.

The term -x is negative because x is positive on the given interval.

The term (x+5) is positive because x is positive on the given interval.

The term (x+1) is positive because x is positive on the given interval.

The term (x-2) is negative because x is between 0 and 2 on this interval.

When you multiply these terms together, you have one negative term (-x) and three positive terms (x+5), (x+1), and (x-2). The overall product is negative because there is an odd number of negative terms.

Therefore, the sign of the function f(x) on the interval 0 < x < 2 is always negative.