At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga.
Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus.
Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.
At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga.
Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus.
Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.
1. If
f '(x) > 0
for all x on (a,b),
then f is
increasing
on [a,b]
2. If
f '(x) < 0
for all x on (a,b),
then f is
decreasing
on [a,b]
3. If
f '(x) = 0
for all x on (a,b),
then f is
constant
on [a,b]
You could graph the first derivative. Or you could set it equal to 0 and solve and then test each interval to see if the derivative is positive or negative within the interval.
That takes us to the second derivative. If the second derivative is positive, the function is concave upward. Downward if negative. Inflection point if 0
Concavity Theorem:
If the function f is twice differentiable at x=c, then the graph of f is concave upward at (cf(c)) if f(c)0 and concave downward if f(c)0.
to find whether a function is increasing or decreasing, you need the first derivative.
Graph the first derivative.
For each x-value:
If the first derivative is positive, the function is increasing at that x-value
If the first derivative is negative, the function is decreasing at that x-value
If the first derivative is zero, the function is neither increasing nor decreasing at that x-value.
to find concavity and inflection points, you need the second derivative.
Graph the second derivative.
for each x-value:
If the second derivative is positive, the function is concave up at that x-value
If the second derivative is negative, the function is concave down at that x-value
If the second derivative is zero (and not constant) the function has a point of inflection at that x-value