Question

Consider the equation below. (if an answer does not exist, enter dne.) f(x) = x4 − 32x2 + 5 (a) find the interval on which f is increasing. (enter your answer using interval notation.)

Answer 1

The function f(x) is increasing on the interval (-4, 0) ∪ (4, ∞).

Answer 2

Answer:

The intervals of increase are $\left(-4, 0\right) \cup \left(4, \infty\right)$

Step-by-step explanation:

When a function is increasing, its derivative is positive. So if we want to find the intervals where a function increases, we differentiate it and find the intervals where its derivative is positive.

Let's find the intervals where $f(x) = x^4 -32x^2 + 5$ is increasing.

First, we differentiate f(x)

$\frac{d}{dx} f = \frac{d}{dx} (x^4 -32x^2 + 5)\\\frac{d}{dx} f = \frac{d}{dx}\left(x^4\right)-\frac{d}{dx}\left(32x^2\right)+\frac{d}{dx}\left(5\right)\\\frac{d}{dx} f =4x^3-64x$

Now we want to find the intervals where $\frac{d}{dx} f$ is positive This is done using critical points, which are the points where $\frac{d}{dx} f$ is either 0 or undefined.

$4x^3-64x=0\\4x\left(x^2-16\right)=0\\4x\left(x+4\right)\left(x-4\right)=0\\\\\mathrm{The\:solutions\:are}\\x=0,\:x=-4,\:x=4$

These points divide the number line into four intervals

$(-\infty,-4);(-4,0);(0,4);(4,\infty)$

Let's evaluate $\frac{d}{dx} f$ at each interval to see if it's positive on that interval.

$\left\begin{array}{cccc}Interval&x-value&\frac{d}{dx}f& Verdict\\x$

The intervals of increase are $\left(-4, 0\right) \cup \left(4, \infty\right)$