Extrema + related theorems

Question 1

how to check extrema/critical points

Answer 1

• when f’(x) = 0
• domain endpoint values
• f’(x) does not exist

Question 2

if f’‘(x)>0

Answer 2

concave up

Question 2

mean value theorem

Answer 2

to guarantee there is 1+ x where the tan line = sec line
- f(x) must be differentiable on (a,b)
- f(x) must be continuous on [a,b]

Question 3

if f’‘(x)<0

Answer 3

concave down

Question 4

f’‘(x)=0

Answer 4

• extrema
• f’‘(x) changes sign/changes concavity
• points of inflection

Question 5

second derivative test

Answer 5

if x=a is an extrema and f’‘(a)>0, f(x) concaves up and the extrema is a minimum