Saved Bookmarks
| 1. |
Show that `s in^p theta cos^q theta`attains a maximum, when`theta=tan^(-1)sqrt(p/q)`. |
|
Answer» Let `y = sin^(p) theta cos^(q) theta` Then, `(dy)/(dtheta) = p sin^(p -1)theta cos^(q +1) cos^(q -1)theta sin^(p +1) theta` `= (sin^(p-1)theta cos^(q-1)theta) (p cos^(2) theta - q sin ^(2) theta)` Now, for maxima or minima, we have `(dy)/(d theta) = 0` But `(dy)/(d theta) = 0 rArr sin^( p-1) theta = 0 or cos^(q -1) theta = 0 or p cos^(2) theta - q sin^(2) theta = 0` `rArr theta = 0 or theta = (pi)/(2) or theta = tan^(-1) sqrt((p)/(q))` Moreover, we may write `(dy)/(d theta) = (y (p cos^(2) theta - q sin^(2) theta))/(sin theta cos theta) = y (p cot theta - q tan theta)` `:. (d^(2)y)/(d theta^(2)) = y (-p " cosec"^(2) theta - q sec^(2) theta) + (p cot theta - q tan theta) .(dy)/(d theta)` Thus, at `theta = tan^(-1) sqrt((p)/(q))`, we have `(dy)/(d theta) = 0` and therefore, `[(d^(2) y)/(d theta^(2)) " at " theta = tan^(-1) sqrt((p)/(q))] = sin^(p) theta cos^(q) theta (-p " cosec"^(2) theta - q sec^(2) theta) lt 0` Hence, y is maximum when `theta = tan^(-1) sqrt((p)/(q))` |
|