Question

Determine the critical numbers of the given function and classify each critical point as a relative maximum, a relative minimum, or neither. f(t) =(t^2)/t^2+t-2

Answer 1

$\displaystyle f(t) = \frac{t^2}{t^2+t-2}\\ \text{Differentiating, we obtain}\\ f'(t)=\frac{t^2-4t}{(t^2+t-2)^2}\\ f'(t)=\frac{t^2-4t}{(t^2+t-2)^2}=0\\ \implies t = 0, t= 4\\ \text{Using the 2nd test derivative, we evaluate 2nd order derivative}\\ f''(t) =-\frac{2(t^3-6t^2-4)}{(t-1)^3(t+2)^3}\\ \text{When t = 0, $f''(t) =-1$, using the 2nd derivative test, f is maximum at t = 0}\\ \text{when t = 4, $f''(t) = 0.012$, using 2nd derivative test, f is minimum at t = 4}\\$