PLEASE THANK YOU The graph of f ′ (x), the derivative of f of x, is continuous for all x and consists of five line segments as shown below. Given f (0) = 7, find the absolute minimum value of f (x) over the interval [–3, 0].
(SEE PICTURE ATTACHED)
0
2.5
4.5
11.5

  2.5

Step-by-step explanation:

f'(x) is non-negative, so f(x) is monotonic and increasing. Its minimum value on the interval of interest will be less than f(0) by the amount of the integral of f'(x) from -3 to 0.

That integral is the area of the triangle bounded by the portion of the curve from (-3, 3) to (0, 0) and the x-axis. The area is ...

  A = (1/2)bh = (1/2)(3)(3) = 4.5

So, ...

  f(-3) = f(0) -4.5 = 7 -4.5

  f(-3) = 2.5

_____

The red curve in the attached graph is f(x). Its value at -3 is 2.5.

maximum area = xy = (16)(32) = 512 ft2

to maximize the area of the garden, the side of the fence parallel to the side of the house should be 32 ft long, and the other two sides should both have length 16 ft.'

source