Question

Find the minimum of the function f(x) = (2+sin x)(5 - sinx)

Answer 1

Using calculus techniques?

Answer 2

Completing the square gives

f(x) = -( `sinx - 3/2` )^2 + 10 + (3/2)^2

Not hard to see that the minimum value occurs when |sinx - 3/2| is maximum.
This occurs when sinx = -1.

Answer 3

@ganeshie8 I understand the square completion part, but I don't understand the following steps

Answer 4

\(\color{blue}{\text{Originally Posted by}}\) @ganeshie8
Completing the square gives

f(x) = -( `sinx - 3/2` )^2 + 10 + (3/2)^2

Not hard to see that the minimum value occurs when |sinx - 3/2| is maximum.
This occurs when sinx = -1.
\(\color{blue}{\text{End of Quote}}\)

Answer 5

Consider the expression \(-(x-a)^2 + b\).

What do you think is the maximum value of it ?

Answer 6

Key thing to observe here is that "square" of a real number is never "negative".
Therefore `-(x-a)^2` is never positive.

What is the maximum value of "non positive" numbers ?

Answer 7

@ganeshie8 the maximum value is when (x-a)^2 will result in 0, so the expression would just be "b" . Analogously, f(x) = -(sinx-3/2)^2 + 10 + (3/2)^2 will have it's minimum in the highest value that (sinx-3/2)^2 can assume, and for this case it's sin x = -1 ?

Answer 8

You get how it turns to that from completing the square
f(x) = -( sinx - 3/2 )^2 + 10 + (3/2)^2
\[\large f(x)=-(\sin(x)-\frac{ 3 }{ 2 })^2+\frac{ 49 }{ 4 }\]

What is the minimum possible value for f(x)?
To get the smallest value for f(x), you want the smallest value from that squared term so you get the biggest negative value out from that term. -(sin x - 3/2)^2

Remember the sine function is limited to values -1 to 1. So the largest that sin x-3/2 can be is when sin x is -1. To give -(-1-3/2)^2
-(-5/2)^2 or -25/4

Answer 9

Sorry i meant to say, you want that (sin x -3/2) to be the largest value either + or - since it is squared to become positive anyway, and the minus sign is out front of the term. Here the largest happens to be the negative value when sin(x) is -1.

Answer 10

Also if you want to just look at the expanded function
\[\large f(x)=10+3\sin(x)-(\sin(x))^2\]

you can see how the smallest value is for the sin(x) being -1.
or for this form
\[\large f(x)=(\cos(x))^2+3\sin(x)+9\]

reason for sin x=-1 while cos(x)=0

Answer 11

Clever! Your method looks a lot nicer @DanJS

Answer 12

@Styxer looks you have it! Let me know if something is not clear...

Answer 13

@DanJS and @ganeshie8 thanks for your help!! Trigonometric functions are very unique, I'll be more careful when solving these kind of exercises.