15. If f(x) = min(3x + 2, 1 - 5x), what is the maximum possible value of f(x)? (a) 7/8 (b) 8/7 (c) 13/8 (d) 7/13
Answer (1)
To find the maximum possible value of the function f ( x ) = min ( 3 x + 2 , 1 − 5 x ) , we need to determine when each part of the expression inside the minimum function is largest. The function min ( a , b ) selects the smaller value between a and b .
Finding the Intersection Point:
To find when 3 x + 2 = 1 − 5 x , set the expressions equal to each other:
3 x + 2 = 1 − 5 x
Solve for x :
3 x + 5 x = 1 − 2
8 x = − 1 x = − 8 1
Evaluate f ( x ) Near Intersection Point:
At x = − 8 1 , calculate both expressions:
For 3 x + 2 :
3 ( − 8 1 ) + 2 = − 8 3 + 8 16 = 8 13
For 1 − 5 x :
1 − 5 ( − 8 1 ) = 1 + 8 5 = 8 13
Since both expressions give us 8 13 at x = − 8 1 , and we need the minimum, f ( x ) = 8 13 .
Determine the Maximum Possible Value:
Since the function f ( x ) takes the minimum value of the two expressions and they intersect at 8 13 , this intersection point gives the highest possible value of the minimum function.
Therefore, the maximum possible value of f ( x ) is 8 13 .
The correct option is (c) 8 13 .
