find the maximum and/or minimum and state whether it is a relative or absolute maximum/minimum. estimate if…

find the maximum and/or minimum and state whether it is a relative or absolute maximum/minimum. estimate if necessary. (8 points)\n3)\n\n\n\nsketch the graph of the piecewise function. (20 points)\n4) $f(x)=\\begin{cases}4x, & x < 0 \\\\ 2x + 1, & x > 1\\end{cases}$\n\n\n

find the maximum and/or minimum and state whether it is a relative or absolute maximum/minimum. estimate if necessary. (8 points)\n3)\n\n\n\nsketch the graph of the piecewise function. (20 points)\n4) $f(x)=\\begin{cases}4x, & x < 0 \\\\ 2x + 1, & x > 1\\end{cases}$\n\n\n

Answer

Explanation:

Step1: Analyze the first part of the piece - wise function ($x < 0$)

For the function (y = 4x) when (x<0).

  • Choose some values of (x) less than (0). For example, if (x=- 1), then (y = 4\times(-1)=-4); if (x = - 2), then (y=4\times(-2)=-8).
  • The function (y = 4x) is a straight - line with a slope (m = 4) and (y) - intercept (b = 0). Since (x<0), we draw the part of the line (y = 4x) to the left of the (y) - axis (excluding the origin because (x\neq0) in this part of the piece - wise function).

Step2: Analyze the second part of the piece - wise function ($x>1$)

For the function (y=2x + 1) when (x>1).

  • Choose some values of (x) greater than (1). For example, if (x = 2), then (y=2\times2+1=5); if (x=3), then (y=2\times3 + 1=7).
  • The function (y=2x + 1) is a straight - line with a slope (m = 2) and (y) - intercept (b = 1). Since (x>1), we draw the part of the line (y=2x + 1) to the right of (x = 1) (excluding the point (x = 1)).

Answer:

To sketch the graph:

  1. For (y = 4x,x<0): Plot points like ((-1,-4),(-2,-8)) and draw a straight line (dashed at (x = 0) since (x\neq0) for this part) to the left of the (y) - axis.
  2. For (y=2x + 1,x>1): Plot points like ((2,5),(3,7)) and draw a straight line (dashed at (x = 1) since (x\neq1) for this part) to the right of (x = 1).