Determine all the points of local maxima and local minima of the following function: f(x) = (-¾)x4 – 8x3 – (45/2)x2 + 105

Dear Student

Given function: f(x) = (-¾)x^4– 8x^3– (45/2)x^2+ 105

Thus, differentiate the function with respect to x, we get

f ′ (x) = –3x^3– 24x^2– 45x

Now 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 = 0

Now, again differentiate the function:

f ″(x) = –9x^2– 48x – 45

Take -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 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.

Thanks