This answer presumes that only one statement is untrue, as you did not include a line about "Circle all that apply."
In order to determine things such as whether the function is increasing or decreasing on a given interval, or its concavity, we must take the derivative of the function.
Since we have e^(1/x)e1x, we must use the chain rule, which states that given f(x) = g(h(x)), (df)/dx = (dh)/dx (dg)/(dh)f(x)=g(h(x)),dfdx=dhdxdgdh. With g(h) = e^h, h(x) = 1/x, (dh)/(dx) = -1/x^2, (dg)/(dh) = e^hg(h)=eh,h(x)=1x,dhdx=−1x2,dgdh=eh, so...
(df)/dx = -1/(x^2)e^(1/x)dfdx=−1x2e1x
Looking at this function, from the interval (-1/2, 0)(−12,0) our x-value will be negative. With this being the case, some analysis shows us that the e^(1/x)e1x component is greater than 0 for all x, by the definition of an exponential function. Further, x^2>=0x2≥0 for all x, so the denominator for -1/x^2−1x2 is positive for all x, meaning that -1/x^2<0−1x2<0 for all x. That means that f'(x) <0 for all x, which in turn means the function is always decreasing. Thus, it is not increasing on (-1/2, 0), and thus 3 is false.
