Question

What is the minimum value of the function g(x)= 1/x * sin (pi *x) on the interval [1,2]?

I got -2/3. Is this right? thanks.

Answer 1

Is the function
f(x)=1/(x sin(πx))
or
f(x)=(1/x) sin(πx) [ as written ]?

Answer 2

On a closer look, the first equation f(x)=1/(x sin(πx)) is undefined at x=1 and x=2, so probably not what you want.

For equation 2,
f(x)=(1/x) sin(πx)
you would find
f'(x)= (%pi*cos(πx))/x - sin(πx)/x^2
equate f'(x) to zero to find the minimum, and solve for a root between x=1 and 2.

It seems that numerical method (such as Newton's method or bisection method) will be required to find the root, which is situated at x=1.43. Find f"(x) to verify that f"(1.43)>0 for a minimum. Else, you could do the check using a graphing calculator.

Answer 3

Note:
-3/2 is not in [1,2], so it cannot be the correction solution.