UPSC NDA Math Model Paper 4

For the given function f(x)=3x4+4x3−12x2+12, first let's find the points of local maxima or minima:
f′(x)=12x3+12x2−24x=0
⇒12x(x2+x−2)=0
⇒x(x+2)(x−1)=0
⇒x=0 OR x=−2 OR x=1.
f"(x)=36x2+24x−24
f"(0)=36(0)2+24(0)−24=−24
f"(−2)=36(−2)2+24(−2)−24 =144−48−24=72
f"(1)=36(1)2+24(1)−24 =36+24−24=36
Since, at x = 0 the value f′′(0)=−24<0 , the local maximum value of the function occurs at x = 0