Find all the points of local maxima and local minima of the functionf(

1.	Find all the points of local maxima and local minima of the functionf(x) = -(3/4)x4 - 8x3 - (45/2)x2 + 105
Answer» f ′ (x) = –3x³ – 24x² – 45x = – 3x (x² + 8x + 15) = – 3x (x + 5) (x + 3) f ′ (x) = 0 ⇒ x = –5, x = –3, x = 0 f ″(x) = –9x² – 48x – 45 = –3 (3x² + 16x + 15) f ″(0) = – 45 < 0. Therefore, x = 0 is point of local maxima f ″(–3) = 18 > 0. Therefore, x = –3 is point of local minima f ″(–5) = –30 < 0. Therefore x = –5 is point of local maxima.

Find all the points of local maxima and local minima of the functionf(x) = -(3/4)x4 - 8x3 - (45/2)x2 + 105

Answer»

f ′ (x) = –3x³ – 24x² – 45x

= – 3x (x² + 8x + 15) = – 3x (x + 5) (x + 3)

f ′ (x) = 0 ⇒ x = –5, x = –3, x = 0

f ″(x) = –9x² – 48x – 45

= –3 (3x² + 16x + 15)

f ″(0) = – 45 < 0. Therefore, x = 0 is point of local maxima

f ″(–3) = 18 > 0. Therefore, x = –3 is point of local minima

f ″(–5) = –30 < 0. Therefore x = –5 is point of local maxima.