Question

How do I find the interval on which a function is increasing?

Answer 1

1) look at the graph

2) take derivative and see where it is positive

Answer 2

\[f(x)=4x^2-40x+101\]
that is the function that I have. The vertex is (5,1)

Answer 3

I haven't graphed it. I should be able to determine this algebraically, correct?

Answer 4

1) find the stationary points (i.e. where the derivative is zero)
2) check if the derivative is positive or negative between each of these points to determine if its increasing or decreasing

Answer 5

What do you mean by the "derivative?"

Answer 6

This is parabola. It is concave up since it is positive x^2.

find the vertex

Answer 7

vertex is (5,1)

Answer 8

f'(x) is the derivative of f(x)

Answer 9

concave up ---> min

Answer 10

so it would increase after vertex point

Answer 11

vertex is (5,1) so decreasing on
\[(-\infty, 5)\] increasing on
\[(5,\infty)\]

Answer 12

in your case:\[
f'(x)=8x-40\]

Answer 13

you can do this before you know calculus. parabola looks like this

Answer 14

so the range is [1,infinity) and the interval is (5,infinity)?

Answer 15

range is
\[[1,\infty)\] if your vertex is right

Answer 16

the vertex is correct.

Answer 17

what exactly is the interval?

Answer 18

not the value, but the definition.

Answer 19

range you are looking along the y - axis, but when asked for intervals of increase and decrease you are referring to the x - axis, i.e. the inputs. that is why you say
"decreasing on
\[(-\infty,5)\]