Question

yIP is paralle FaPat P df be the point at which f has a local extreme value. Then, dt t-t Let C be expressed parametrically as rit) (x() yit, and let P(ab) (())) This means that Viab)-r (6)Vg(a b) Because r(t) is orthogonal to C. VP) is orthogonal to the line tangent to C at P The gradient is maximized VgP) is orthogonal to C at P because gixy)0 is a level curve Therefore, the two gradients are parallel Explain why, at a point that maximizes or minimizes f subject to a constraint g(x.y) 0, the gradient of f is parallel to the gradient of g. Use the diagram to the right fatains its manimem Level curver value on the consainn curve at FeiP) -3 Coestrain curve Tangent to Cat P Cgs)-0 P pralle to TePP df be the point at which f has a local extreme value. Then, It-to Let C be expressed parametrically as r(t) (x(t) y(). and let P(a.b) (x (to) y (o)) dt Vg(a.b) Because r'(t) is orthogonal to C, Vf(P) is orthogonal to the line tangent to C at P The gradient This means that Vab)-r' (o) is maximized O Thorefore. the two gradients are parallel Vg(P) is orthogonal to C at P because 9xy)-0 is a level curve Let C be expressed parametrically as r(t) (x(t) y(t)), and le This means that Vf(a.b) r (to) = Vg(a is maximized it P because g(x.y) 0 is a level c is equal to zero. is maximized Click to select your answer(s) Save for Later olain why, at a point that maximizes or minimizes f subject to a constraint c dient of g. Use the diagram to the right Let C be expressed parametrically as rt)= (x(t) y(t)), and let P(a,b) (x (to) This means that Vf(a b) r' (to) = Vg(a b) Because r (t) is maximized Vg(P) is orthogonal to C at P becau erefore Vf(a b) r (to) = Vg(a,b). Vfía b) r (to) 0 Click to select your answer(s). rt) (x) y(0) et C be expressed parametrically and let P(a b) (x(0) ¥ ('o )) be the point at which f has a local extreme value. The as This means that Viab).r (to) Vg(a.b) | Because r (t) is orthogonal to C. Vf(P) is orthogonal to the line tangent is maximized gP) is orthogonal to C at P because gxy) 0 is a level curve Therefore, t parallel orthogonal tangent Click to select your answer(s) df be the point at which f has a local extreme value. Then, dt t-to y) and let P(a.b) = (x(to) y (to)) (0)= Vg(a.b). Because r'(t) is orthogonal to C. VP) is orthogonal to the line tangent to C at P. The gradient Therefore, the two gradients are paralle v) 0 is a level curve tangent orthogonal (*()y(0) be rt) (x(t) y(t)), and let P(a,b) Let C be expressed parametrically as This means that Vf(a.b) r (to) = Vg(a,b). Because r (t) is orthog is maximized Vg(P) is orthogonal to C at P because g(xy= 0 is a level curve Therefore, the two g g(x.y) = 0 is a level curve Vfa.b) r (to) Vg(a b). Click to select your answer(s)

Answer 1

From the diagram,

Then, f t t=to is egual to gefo Ans f has a local extreme value at |t=to ] This means hatf(a,b).r'(to) Y(t) is Because tangent tangent Orthogonal to c.Vf(P) is (orthogonal to the line The to C at P. (is gradient P Cat to because is evel There fore the gradients are parallel- Curve