Question

For the function y=-1+6 cos(2pi/7(x-5)) what is the minimum value?

Answer 1

Usually one will differentiate the function to find the minimum/maximum point, but in this case differentiating yields:

`(2\pi)/(7(x-5)^(2))\sin{(2\pi)/(7(x-5))}}`
which contains multiple solution if one tries to solve for x when the differentiated form is 0.

I would, though, venture a guess that the minimum value would be (approaching) 5, since the function would be undefined in the vicinity.

If, however, the function is

`-1+\cos{(2\pi)/(7)(x-5)}}`
Then differentiating and equating to 0 yields:

`\sin{(2\pi)/(7)(x-5)}}=0`
which gives:

`x=5` or
`8.5`

We reject x=5 as it is when it ix the maximum and thus,

`x=8.5\pm7n`, for
`n=0,\pm 1,\pm 2, ...`

Answer 2

The value of y lie in the interval [-7,5] and the minimum value of given function is -7.

Explanation:

The given function is

`y=-1+6\cos ((2\pi)/(7)(x-5))`

We know that the value of cosine function lies between -1 and 1.

By the above property of cosine function we get

`-1\leq \cos ((2\pi)/(7)(x-5))\leq 1`

Multiply 6 each side.

`-6\leq 6\cos ((2\pi)/(7)(x-5))\leq 6`

Subtract 1 from each side.

`-6-1\leq 6\cos ((2\pi)/(7)(x-5))-1\leq 6-1`

`-7\leq y\leq 5`
`[\because y=-1+6\cos ((2\pi)/(7)(x-5))]`

Therefore the value of y lie in the interval [-7,5] and the minimum value of given function is -7.