Finding Critical Points


In Exercises 41–50, determine all critical points and all domain endpoints for each function.


y = x² − 32√x

Use ƒ' and ƒ" to complete parts (a) and (b). 

a. Find the intervals on which f is increasing and the intervals on which it is decreasing.

b. Find the intervals on which f is concave up and the intervals on which it is concave down.

ƒ(x) = x√(x +9)

{Use of Tech} Every second counts You must get from a point P on the straight shore of a lake to a stranded swimmer who is 50 from a point Q on the shore that is 50 m from you (see figure). Assuming that you can swim at a speed of 2 m/s and run at a speed of 4 m/s, the goal of this exercise is to determine the point along the shore, x meters from Q, where you should stop running and start swimming to reach the swimmer in the minimum time. <IMAGE>

b. Find the critical point of T on (0, 50).

Finding Critical Points

In Exercises 41–50, determine all critical points and all domain endpoints for each function.

f(x) = x(4 − x)³

Finding Critical Points

In Exercises 41–50, determine all critical points and all domain endpoints for each function.

y = x² + 2/x

Theory and Examples

In Exercises 51 and 52, give reasons for your answers.

Let f(x) = (x − 2)²ᐟ³.

a. Does f′(2) exist?

Theory and Examples

In Exercises 51 and 52, give reasons for your answers.

Let f(x) = |x³ − 9x|.

b. Does f'(3) exist?

Theory and Examples

In Exercises 53 and 54, show that the function has neither an absolute minimum nor an absolute maximum on its natural domain.

y = x¹¹ + x³ + x − 5

Theory and Examples

Maximum height of a vertically moving body The height of a body moving vertically is given by s = −12gt² + υ₀t + s₀,  g > 0, with s in meters and t in seconds. Find the body’s maximum height.