question_answer 67)

Find the  local maxima and local minima, if any, of the following functions. Find also the local maximum and the local minimum values, as the case may be : (I)     f(x) = x2 (II)    g(x) = x3 ? 3x (III)   h(x) = sin x + cos x, 0 < x < (IV)  f(x) = sin x ? cos x, 0 < x < (V)   f(x) = x3 ? 6x2 + 9x + 15 (VI)  g(x) (VII) (VII) f(x) = x  

Answer:

f(x) = x2             For critical points f?(x) = 0             Now f?(0) = 2              has local minima at x = 0 and local minimum value of f(x) is f(0) = 0. II.    g(x) = x3 ?3x                                     Now g??(x) = 6x       g?(x)|x= ?1 = ?6 < 0       g?(x)|x=1 = 6 > 0        g(x) has local maxima at x = ?1       and local minimum at x = 1       Local max. value = g(?1) = ?1 + 3 = 2       Local min. value = g(1) = 1?3 = ?2 III.   h(x) = sin x + cos x in             h?(x) = 0             Now h?(x) = ? (sin x + cos x)       h?(x) = ? (sin x + cos x)              has local maxima at and local max. value       IV.  f(x) = sin x ? cos x in (0, )             f?(x) = 0                   Now                    has local maxima at       and local minima at        Local max. value             and local min. value =       V.   f(x) = x3 ? 6x2 + 9x + 15                         Now f?(x) = 6x ? 12 = 6(x ? 2)       f?(x)|x=1 = ?6 < 0       f?(x)|x = 3 = 6 > 0       f(x) has local maxima at x = 1       and local minima at x = 3       Local max. Value = f(1)       = 1 ? 6 + 9 + 15 = 19                             and local minimum value = f(3)       = 27 ? 54 + 27 + 15 = 15 VI.              g?(x) = 0                                     Since x > 0, therefore taking x = 2                                 Now g?(x) = 0 +                          has local minima at x = 2 and local min.                   VII.                                     g?(x) = 0             Now g? (x) = ? 2                                has local maxima at x = 0 and local maximum value = g(0) = VIII.                   f?(x) = 0       Now f?(x) =                                      has local maxima at       Local maximum value