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.