Would it be possible to help me with this other problem Than

Would it be possible to help me with this other problem? Thank you

Let f(x,y,z) = x2y + y2z. Evaluate f. Find an equation of the tangent plane to the surface x2y + y2z = - 1 at the point (2,-1,3). Find a unit vector v such that / (x, y, z) has a negative rate of change at (2,-1,3) in the direction v.

Solution

(a) grad (f)= fx i +fy j+fz k= 2xy i + (x²+2yz) j +y² k

(b) equation of tangent plane

z-z0= fx(x0,y0,z0)*(x-x0) +fy(x0,y0,z0)*(y-y0)

=>z-3=-4*(x-2) + (-2)*(y+1)

=>z+4x+2y=9

(c)unit vector in direction opposite to gradient = -(-4i - 2j +k)/√21=(4i+2j-k)/√21