The function f is defined above. Which of the following equivalent for

Question

f(x) = -x²+6x+20

The function f is defined above. Which of the following equivalent forms of f(x) displays the maximum value of f as a constant or coefficient?

(A) f(x) = (x-3)²+11

(B) f(x) = -(x-3)²+29

(C) f(x) = -(x+3)² + 11

(D) f(x) = (x+3)² +29

(E) f(x) = -(x-3)²-11

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imorenor · Answer

Why is option B the correct answer?

f(x) = (x-3)²+29
f(x) = (x²-6x+9)+29
f(x) = x²-6x+38 <>-x²+6x+20

shouldn't it be the correct answer?
f(x) = -(x-3)²+29
f(x) = -(x²-6x+9)+29
f(x) = -x²+6x-9+29
f(x) = -x²+6x+20

Kinshook · Answer

Given: f(x) = -x²+6x+20

The function f is defined above.

Asked: Which of the following equivalent forms of f(x) displays the maximum value of f as a constant or coefficient?

f(x) = -x²+6x+20 = - (x^2 - 6x + 9) +9 + 20 = -(x-3)^2 + 29

IMO B

RenB · Answer

I didnt quite understand the qs. Can someone pls help?

Flower_Child · Answer

The question wants us to convert the given equation in a form in which the MAX value of the function is visible .

But we don't know what is the MAX value. To find that we have to convert the existing equation in some (a+b)^2 or (a-b)^2 form as shown below:

f(x) = -x²+6x+20
f(x) = -x²+6x-9+29
f(x) = -(x²-6x+9)+29
f(x) = -(x-3)²+29 --> Now we can see this will give max value when the constant (29) is not subtracted by any number . The MAX value is 29.
We see that this equation already has the max value in it. So this is our required answer.

Correct Option -  (B)

Simsim34 · Answer

Step 1: Choosing a Test Value  
 A quick way to identify the correct answer in a multiple-choice setting is to substitute a test value into the original function and compare the results with the given options. 
 
Let's choose x=1x=1 as our test value and compute f(1)  :

f(1)=−(1)^2+6(1)+20 =   −1+6+20   = 25

Step 2: Evaluating Each Answer Choice  
 Now, we check which option gives f(1)=25

(A)  (1−3)^2+11=(−2)^2+11=4+11=15 ❌  
   (B)  −(1−3)^2+29=−4+29 = 25 ✅  
   (C)  −(1+3)^2+11=−(4)^2+11=−16+11=−5 ❌  
   (D)  (1+3)^2+29=(4)^2+29=16+29=45 ❌  
   (E)  −(1−3)^2−11=−4−11=−15 ❌
   
 The only correct answer is  (B) , as it produces the expected result.

PS : You can do with all numbers and get only one good answer except 0 in which you will have A and B good but we need to find the value that maximise so B will be right answer.

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The function f is defined above. Which of the following equivalent for

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