lock in for calc BC test 10/2/25

When is c a critical number?

When f’(c) is zero or undefined

When is f(x) increasing?

When f'(x) > 0

When is f(x) decreasing?

When f’(x) < 0

f’(c) = 0 or is undefined, and f’(x) changes from + to - at x = c

(c, f(c)) is a relative max

f’(c) = 0 or is undefined, and f’(x) changes from - to + at x = c

(c, f(c)) is a relative min

c is a POI when

f’’(c) = 0 or is undefined, and f’’(c) changes signs at x = c

f(x) is concave up when

f’’(x) > 0

f(x) is concave down when

f’’(x) < 0

(c, f(c)) is a relative min when

f’(c) = 0 or is undefined, and f’’(c) > 0

(c, f(c)) is a relative max when

f’(c) = 0 or is undefined, and f’’(c) < 0

f is continuous, and f(a) does not equal f(b)

Therefore f(c) = k for a < c < b

Intermediate Value Theorem

f is continuous on [a, b]

There must be an absolute min and max on [a, b]

Extreme Value Theorem

f is continuous on [a, b] and f is differentiable on (a, b)

There is a c in (a, b) where f(b)-f(a)/b-a = f’(c)

Mean Value Theorem

f is continuous on [a, b], f is differentiable on (a, b), and f(a) = f(b)

There exists a c in (a, b) where f’(c) = 0

Rolle’s Theorem (type of MVT)