The polynomial Px4 x3 3 x2 2 x has local maxima and minima

The polynomial P(x)=4 x^3 + 3 x^2 - 2 x has......................? local maxima and minima.

Solution : polynomial

P(x)=4x^3+3x^2-2x

Differentiate this polynomial w.r.t. x , to get

P\'(x)=12x^2+6x-2

For maxima or minima, P\'(x)=0

12x^2+6x-2=0

6x^2+3x-1=0

x=(-3+-sqrt(9+24))/12

Or x=(-3+-sqrt (30))/12

Or. x=(-3+30)/12, (-3-30)/12

Again differentiate, to get P\"(x)=24x+6.

For x=(-3+30)/12

P\"(x)=24(-3+30)/12 + 6

=-6+30+6

=+ vie

Therefore at x = (-3+30)/12 the polynomial is local minima.

Similarly, the point x=( -3-30)/12 is local maxima.