Respuesta :

Usually one will differentiate the function to find the minimum/maximum point, but in this case differentiating yields:
[tex]\frac{2\pi}{7(x-5)^{2}}\sin{\frac{2\pi}{7(x-5)}}}[/tex]
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
[tex]-1+\cos{\frac{2\pi}{7}(x-5)}}[/tex]
Then differentiating and equating to 0 yields:
[tex]\sin{\frac{2\pi}{7}(x-5)}}=0[/tex]
which gives:
[tex]x=5[/tex] or [tex]8.5[/tex]

We reject x=5 as it is when it ix the maximum and thus,
[tex]x=8.5\pm7n[/tex], for [tex]n=0,\pm 1,\pm 2, ...[/tex]

Answer:

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

Step-by-step explanation:

The given function is

[tex]y=-1+6\cos (\frac{2\pi}{7}(x-5))[/tex]

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

By the above property of cosine function we get

[tex]-1\leq \cos (\frac{2\pi}{7}(x-5))\leq 1[/tex]

Multiply 6 each side.

[tex]-6\leq 6\cos (\frac{2\pi}{7}(x-5))\leq 6[/tex]

Subtract 1 from each side.

[tex]-6-1\leq 6\cos (\frac{2\pi}{7}(x-5))-1\leq 6-1[/tex]

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

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

Otras preguntas