06 Applications of Derivatives Free Online FlashCards

What is a critical number?

A critical number is a domain value c where f'(c)=0 or f'(c) does not exist.

Does every critical number give a local extremum?

No. A critical number need not be a local extremum; f(x)=x^3 has f'(0)=0 but keeps increasing through x=0.

Why check endpoints on a closed interval?

For an absolute-extrema problem on [a,b], evaluate f(a) and f(b) as well as interior critical numbers.

What hypotheses are required by the Mean Value Theorem?

The function must be continuous on [a,b] and differentiable on (a,b).

For x² on [1,3], what MVT point c results?

For f(x)=x^2 on [1,3], the average rate is 4, and solving f'(c)=4 gives c=2.

How does derivative sign determine monotonicity?

If f'(x)>0 throughout an interval, f is increasing there; if f'(x)<0 throughout it, f is decreasing there.

What does the First Derivative Test classify?

A change from positive to negative in f' gives a local maximum; a change from negative to positive gives a local minimum.

What does the sign of f'' indicate?

If f''(x)>0, the graph is concave up and its slopes are increasing; if f''(x)<0, it is concave down and its slopes are decreasing.

What condition is necessary for an inflection point?

An inflection point requires an actual change in concavity. Having f''(c)=0 or f''(c) undefined only identifies a possible location.

How does the Second Derivative Test classify a critical point?

When f'(c)=0 and f'' is continuous near c, f''(c)>0 gives a local minimum and f''(c)<0 gives a local maximum.

What if f''(c)=0 in the Second Derivative Test?

If f'(c)=0 and f''(c)=0, the Second Derivative Test is inconclusive; use the First Derivative Test or another method.

What are the critical numbers of x³−3x²−9x+5?

For f(x)=x³−3x²−9x+5, the critical numbers are x=−1 and x=3.