If y=x3logloge(1+x), then y''(0) equals
0
-1
6loge2
6
Explanation for the correct option.
Step 1: Find first derivative.
y=x3logloge(1+x)y'=3x2loglog(1+x)+x31log(1+x)·11+x·1=x23loglog(1+x)+x1+xlog(1+x)
Step 2: Find second derivative.
y''=2x3loglog(1+x)+x1+xlog(1+x)+x23log(1+x)(1+x)+1+xlog(1+x)-x1+x+log(1+x)1+x1+xlog(1+x)2
Now,
y''0=203loglog(1+0)+01+0log(1+0)+023log(1+0)(1+0)+1+0log(1+0)-01+0+log(1+0)1+01+0log(1+0)2=0
Hence, option A is correct.
If x+y=-2, then x3+y3+8 equals
if x/y+y/x = -1 (x,y is not equal to 0 ) then x3-y3 = ?