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Determine all the points of local maxima and local minima of the following function: f(x) = (-¾)x 4 – 8x 3 – (45/2)x 2 + 105

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one month ago Anand Kumar Pandey
1677 Points
```							Dear StudentGiven function: f(x) = (-¾)x^4– 8x^3– (45/2)x^2+ 105Thus, differentiate the function with respect to x, we getf ′ (x) = –3x^3– 24x^2– 45xNow take, -3x as common:= – 3x (x^2+ 8x + 15)Factorise the expression inside the bracket, then we have:= – 3x (x +5)(x+3)f ′ (x) = 0⇒ x = –5, x = –3, x = 0Now, again differentiate the function:f ″(x) = –9x^2– 48x – 45Take -3 outside,= –3 (3x^2+ 16x + 15)Now, substitue the value of x in the second derivative function.f ″(0) = – 45 < 0. Hence, x = 0 is point of local maximaf ″(–3) = 18 > 0. Hence, x = –3 is point of local minimaf ″(–5) = –30 < 0. Hence, x = –5 is point of local maxima.Thanks
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one month ago
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