Question

Maxima and minima of
у - 3x^4 + 16x^3+18x^2+ 20​

Answer 1

Step-by-step explanation:

Let us consider the minimum of f(x)f(x) as kk. Then f(x)−kf(x)−k has at least one repeated root. That is

f(x)−k=3x4−16x3+18x2+5−k=3(x−a)2(x2+bx+c)f(x)−k=3x4−16x3+18x2+5−k=3(x−a)2(x2+bx+c)

Expand and equate coefficients, you get k=5−3a2ck=5−3a2c, and

3(b−2a)=−163(c+a2−2ab)=18a(ab−2c)=03(b−2a)=−163(c+a2−2ab)=18a(ab−2c)=0

From third equation if a=0a=0, then k=5k=5. For a≠0a≠0 combinig three equations gives a2=4a−3a2=4a−3. Thus a=1,3a=1,3.

a=1