Find Values of b

Section 2.1
77 & 78


Exercises 77 and 78, find the values
of b such that the function has the given maximum or minimum value.

77. f(x) = −x^2 + bx − 75; Maximum value: 25
78. f(x) = x^2 + bx − 25; Minimum value: −50

Seeking hints and a set up only.

 

hint for : write equation in vertex form
recall Maximum value is y-coordinate of the vertex which is given and it is 25
so, k=25
same for Minimum value

 

Ok. I'll work on this when time allows. Heading to work now.

 

Question 77

y = a(x - h)^2 + k

-x^2 + bx - 75

-(x^2 - bx + 75)

-(x^2 - bx + b^2/4) - b^2/4 + 75

-(x - b/2)(x - b/2) - b^2/4 + 75

-(x - b/2)^2 -b^2/4 + 74

Stuck here.

1. What did I do wrong?

2. Can you complete 77? I will then show my work for 78.

Thanks.

 

y = a(x - h)^2 + k

-x^2 + bx - 75..........group first two terms
(-x^2 + bx) - 75.....factor out -1
-1(x^2 -bx) - 75
-(x^2 -bx+(b/2)^2)-(-(b/2)^2 - 75
-(x -b/2)^2+b^2/4 - 75

-> as you can see k=b^2/4 - 75

since given k=25, we have
b^2/4 - 75=25
b^2/4 =25+75
b^2/4 =100
b^2 =400
b=sqrt(400)
b=20

so, your equation is -x^2 + 20x - 75
or in vertex form:
-(x -20/2)^2+20^2/4 - 75
-(x -10)^2+400/4 - 75
-(x -10)^2+100 - 75
-(x -10)^2+25 -> which confirms that k=25

 

You really are amazing. Very talented. Very smart. You taught Junior High School mathematics, right? At the same time, you can do precalculus, single variable and multivariable calculus and more. You are truly gifted. God has given you this talent to help others. I truly thank you for your help, guidance, and friendship.

 

thank you for the kind words. Your kind words warmed my heart.