Determine all the points of local maxima and local minima of the following function: f(x) = (-¾)x4 – 8x3 – (45/2)x2 + 105
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2)x2 + 105
Determine all the points of local maxima and local minima of the following function: f(x) = (-¾)x4 – 8x3 – (45/2)x2 + 105
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Answer:
Given function: f(x) = (-¾)x4 – 8x3 – (45/2)x2 + 105
Thus, differentiate the function with respect to x, we get
f ′ (x) = –3x3 – 24x2 – 45x
Now take, -3x as common:
= – 3x (x2 + 8x + 15)
Factorise the expression inside the bracket, then we have:
= – 3x (x +5)(x+3)
f ′ (x) = 0
⇒ x = –5, x = –3, x = 0
Now, again differentiate the function:
f ″(x) = –9x2 – 48x – 45
Take -3 outside,
= –3 (3x2 + 16x + 15)
Now, substitue the value of x in the second derivative function.
f ″(0) = – 45 < 0. Hence, x = 0 is point of local maxima
f ″(–3) = 18 > 0. Hence, x = –3 is point of local minima
f ″(–5) = –30 < 0. Hence, x = –5 is point of local maxima.