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givi [52]
2 years ago
11

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.)

Mathematics
2 answers:
ivann1987 [24]2 years ago
6 0
The function f(x) is increasing on the interval (-4, 0) ∪ (4, ∞).

Paul [167]2 years ago
6 0

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)

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