Question

Find the intervals of increase and decrease for the following functions: (a) f(x) = x 3 + 4x + 1

(b) f(x) = x 3 (5 − x) 2 .

(c) f(x) = x + sin x.

(d) f(x) = (x 2 − 4).

(e) f(x) = 2x 3 − 9x 2 + 12x

(f) f(x) = (x 2 − 4)2

(g) f(x) = x 3 − 2x 2 + x + 5 

Answer 1

To find the intervals of increase/decrease of f(x), we need to find intervals, where its derivative is positive and negative respectively.

(a) $f(x) = x^3 +4x + 1, f'(x) = 3x^2+4 \geqslant 4\:\:\forall x \in \mathbb{R}$ , so f(x) increases at whole real number axis

(b) $f(x) = x^3(5-x)^2$

$f'(x) = 3x^2(5-x)^2 + x^3 \cdot2(5-x) \cdot (-1) = \\ = x^2(x-5) \cdot (3(x-5)+2x) = x^2(x-5)5(x-3)$ , and f'(x) sign is the same that sign of $(x-3)(x-5) :$ it's $< 0$ (f(x) decreases) for $x \in (3, 5)$ , it's $>0$ (f(x) increases) for $x \in (-\infty, 3) \cup(5, +\infty)$

(c) $f(x) = x+\text{sin}x, f'(x) = 1+\text{cos}x \geqslant 0 \:\: \forall x \in \mathbb{R}$ , so f(x) increases in all points except f'(x) nulls : $\{x \in \mathbb{R}: \text{cos}x = -1\} = \{\pi+2\pi n, n \in \mathbb{Z}\}$

(d) $f(x) = x^2-4, f'(x) = 2x$ , and $\forall x \in (-\infty, 0) \: f'(x)<0$ (f(x) decreases), and$\forall x \in (0, +\infty) \: f'(x)>0$ (f(x) increases)

(e) $f(x) = 2x^3-9x^2+12x, f'(x) = 6x^2-18x+12$ , so $f'(x) = 6(x^2-3x+2) = 6(x-1)(x-2)$ and $\forall x \in (1, 2) \: f'(x)<0$ (f(x) decreases), and $\forall x \in (-\infty, 1)\cup(2, +\infty) \: f'(x)>0$ (f(x) increases)

(f) $f(x) = (x^2-4)^2, f'(x) = 2(x^2-4)\cdot2x$ , $f'(x) = 4(x+2)x(x-2)$ , so $\forall x \in (-\infty, -2)\cup(0, 2) \: f'(x)<0$ (f(x) decreases), $\forall x \in (-2, 0)\cup(2, +\infty) \: f'(x)>0$ (f(x) increases)

(g) $f(x) = x^3-2x^2+x+5, f'(x) = 3x^2-4x+1$ , so $f'(x) = (3x-1)(x-1)$ and $\forall x \in (\frac{1}{3}, 1) \: f'(x)<0$ (f(x) decreases), and $\forall x \in (-\infty, \frac{1}{3})\cup(1, +\infty) \: f'(x)>0$(f(x) increases)