القيمه الوسطية والنهايات المحلية

First Derivative Test for Local Extrema
Suppose that c is a critical point of a continuous function ƒ, and that ƒ is
differentiable at every point in some interval containing c except possibly at
c itself. Moving across this interval from left to right,
1. if changes from negative to positive at c, then ƒ has a local minimum at c;
2. if changes from positive to negative at c, then ƒ has a local maximum at c;
3. if does not change sign at c (that is, is positive on both sides of c or
negative on both sides), then ƒ has no local extremum at c.
THEOREM 3—Rolle’s Theorem Suppose that is continuous at every
point of the closed interval [a, b] and differentiable at every point of its interior
(a, b). If then there is at least one number c in (a, b) at which
ƒ?scd = 0.
ƒsad = ƒsbd,
y = ƒsxd
The Mean Value Theorem
THEOREM 4—The Mean Value Theorem Suppose is continuous on a
closed interval [a, b] and differentiable on the interval’s interior (a, b). Then there
is at least one point c in (a, b) at which
(1)
ƒsbd - ƒsad
b - a = ƒ?scd
THEOREM 6—L’Hôpital’s Rule Suppose that that ƒ and g are
differentiable on an open interval I containing a, and that
Then
assuming that the limit on the right side of this equation exists.
lim
x:a
ƒsxd
gsxd = lim
x:a
ƒ?sxd
g?sxd ,
Using L’Hôpital’s Rule
To find
by l’Hôpital’s Rule, continue to differentiate ƒ and g, so long as we still get the
form at But as soon as one or the other of these derivatives is different
from zero at we stop differentiating. L’Hôpital’s Rule does not apply
when either the numerator or denominator has a finite nonzero limit