Locate the critical points of the following functions. Then use the second derivative test to determine whether they correspond to local minima or local maxima or whether the test is inconclusive.

Question

which one

Answer 1

dp/dx=(x²+20)-2x(x-4))/(x²+20)²=0 at an extremum, dp/dx=1/(x²+20)-2x(x-4)/(x²+20)².

That is, x²+20-2x²+8x=0, -x²+8x+20=0=-(x-10)(x+2). The critical points are at x=10 and -2.

d²p/dx²=-2x/(x²+20)²-[(x²+20)²(4x-8)-4x(2x²-8x)(x²+20)]/(x²+20)⁴.

When x=10 this is -20/120²-[120²(32)-40(120)²]/120⁴=-20/120²+8/120²<0 (maximum), and p(10)=6/120=1/20.

When x=-2 it’s 4/24²-[24²(-16)+8(24²)]/24⁴>0 (minimum), and p(-2)=-6/24=-¼.

Max at (10,1/20); min at (-2,-¼). If the critical points are the maximum and minimum then all the answer options are incorrect, the closest being answer a. But the points in this answer option are not even on the curve!