Question

Consider the function below. (If an answer does not exist, enter DNE.) g(x) = 190 + 8x3 + x4 (a) Find the interval of increase. (Enter your answer using interval notation.)

Answer 1

Answer:

The interval of increase of g(x) is $(-6,+\infty)$.

Step-by-step explanation:

The interval of increase occurs when first derivative of given function brings positive values. Let be $g(x) = 190 + 8 \cdot x^{3} + x^{4}$, the first derivative of the function is:

$g'(x) = 24 \cdot x ^{2} + 4\cdot x^{3}$

$g'(x) = 4 \cdot x^{2}\cdot (6+x)$

The following condition must be met to define the interval of increase:

$4\cdot x^{2} \cdot (x+6) > 0$

The first term is always position due to the quadratic form, the second one is a first order polynomial and it is known that positive value is a product of two positive or negative values. Then, the second form must satisfy this:

$x + 6 > 0$

The solution to this inequation is:

$x > - 6$

Now, the solution to this expression in interval notation is: $(-6,+\infty)$