How to find the minimum value of this function

[abel muroi]

I was given this problem...

What is the minimum value of f(x) = (x + 3)² - 5 defined for all real x.


A. x = -5

B. x = -3

C. x = 4

D. Not defined


I'm not sure if i did this correctly, but i used all the answers and plugged them in for x in the original equation, and -3 was the minimum value

x = -3

f(x) = (x + 3)² - 5
f(x) = (-3 + 3)² - 5
f(x) = (0)² - 5
f(x) = 0 - 5
f(x) = -5

This is the method that I used to get my answer, which is B

Is B the right answer, or did I miss something?

 

[Ishuda]

I was given this problem...

What is the minimum value of f(x) = (x + 3)² - 5 defined for all real x.


A. x = -5

B. x = -3

C. x = 4

D. Not defined


I'm not sure if i did this correctly, but i used all the answers and plugged them in for x in the original equation, and -3 was the minimum value

x = -3

f(x) = (x + 3)² - 5
f(x) = (-3 + 3)² - 5
f(x) = (0)² - 5
f(x) = 0 - 5
f(x) = -5

This is the method that I used to get my answer, which is B

Is B the right answer, or did I miss something?

What if you had had a hundred choices or a thousand? What you should note is that the square of a real number is always non-negative and the smallest it can be is zero (for zero squared).

Now for the question: Note that (x + 3)² is the only place x shows up in the definition of f(x) and, since that is non-negative, f(x) is greater than or equal to -5. It is equal to -5 only when x+3 is zero or x=-3. So the minimum occurs at x=-3 and f(-3)=-5

EDIT:You got the right x value but not the right answer [see the bold above]. Actually the correct answer is not offered as a choice [assuming you have stated the problem correctly] so I don't know how you could answer the problem correctly.

 

[pka]

I was given this problem...
What is the minimum value of f(x) = (x + 3)² - 5 defined for all real x.

A. x = -5 B. x = -3 C. x = 4 D. Not defined

Strictly the answer is not there READ the question: What is the minimum value of f(x)?

The answer to that question is \(\displaystyle -5\) that is the minimum value of \(\displaystyle f(x)\).

Had the question said, "For what value is \(\displaystyle f(x)\) minimum?" then B. is correct. But that is not what was asked.