find all local maximum and minimum points by the second derivative test when possible y=2+3×+3 (b) y=6×+sin3×

Step-by-step explanation:

Given a function f, whose derivatives are f' and f'', a value x is a critical point if f'(x) =0. A value x is a minimum of f if it is a critical point and f''(x) >0 and it is maximum if f''(x)<0. We will perfom the following steps:

1. Calculate the derivative f'.

2. Solve f'(x) =0.

3. Determine if the x value found in 2 is a minimum or a maximum using f''.

Recall the following properties of derivatives

where c is a constant.

where f,g are differentiable.

where c is a constant.

(chain rule)

Case 1: f(x) = 2+3x+3.

Using the properties from above, we have

1.

2. The equation f'(x)=0 where f'(x) = 3 has no solution.

3. Based on the previous result, f has no maximum nor minimum.

Case 2:

1.

2. We have the equation

which is equivalent to

Recall that the cosine function only takes values in the set [-1,1]. So, this equation has no solution.

3. Based on the previous result, f has no maximum nor minimum.

34.5 cm

step-by-step explanation:

generally for a circular segment of angle, θ° and radius r

arc length

= (θ/360) x circumference

= (θ/360) x 2πr

in our case, θ = 60°, π=3.142 and r = 33 cm

substitute into above formula

arc length

= (60/360) x 2(3.142)( 33) = 34.5 cm