So I have a question that asks,
Given $f(x) = 7 + 10x ^ 3 - 5x ^ 4 - 2x ^ 5$ calculate the percentage of the interval $[-6, 4]$ on which $f(x)$ is increasing.
So from this I determined that I must differentiate $f(x)$ and determine the intervals for which $f(x)$ is increasing and decreasing. Then determine how many numbers of the interval $[-6, 4]$ $f(x)$ is increasing for and express this number as a ratio to how many numbers are in the interval and then convert this to a percentage.
So the derivative of
$$f(x) = -10x ^ 2(x + 3)(x - 1)$$
The inequality statements for which f(x) is increasing and decreasing are
$$x < - 3, f'(x) < 0$$ $$3 < x < 0, f'(x) > 0$$ $$0 < x < 1, f'(x) > 0$$ $$x > 1, f'(x) < 0$$
so from this I know that there are $2$ numbers for which $f(x)$ is increasing for and there are $11$ numbers in the interval because I'm including the $-6$ and $4$ so
$$\frac{2}{11} = \frac{x}{100} = 18.1818\%$$
So given
$$f(x) = 7 + 10x ^ 3 - 5x ^ 4 - 2x ^ 5$$
$f(x)$ is increasing for $18.1818\%$ of the interval $[-6 , 4]$.
Basically what I want to know is if my understanding of this question and my solution is correct? I have no idea how to check my work on the percentage part.
Your derivative is positive on the intervals $(-3,0)$ and $(0,1)$. This constitutes a total length of $4$ out of the entire length of $10$. Therefore the answer is $40\%$.
Here is a little tool to help answer questions like these (over a single interval):
https://www.desmos.com/calculator/2kxnwwcjh1
The percentage will be $$\dfrac{\text{Area of green rectangle}}{\text{Area of entire rectangle}}$$