Question

Test whether the function f(x) = x³ + 6x² + 12x – 7 is increasing or decreasing for all x ∈ R.

Answer 1

Given function:

f(x) = x³ + 6x² + 12x – 7

Step 1: Find first derivative

f'(x) = 3x² + 12x + 12

Factor it:

f'(x) = 3(x² + 4x + 4)

f'(x) = 3(x + 2)²

Step 2: Check the sign of f′(x)

Since 3(x + 2)² ≥ 0 for all x ∈ R

(x + 2)² is always non-negative

It becomes 0 only at x = –2

Thus, f'(x) ≥ 0 for all x.

Step 3: Conclusion

Since f'(x) ≥ 0 for all real x,

The function is increasing for all x ∈ R.