The direction in which a scalar field( , , )x y z has the largest rate of change atany point with position vectorˆˆ ˆr xi yj zk= + + is the same as that of the vector(A)(B)( )r(C)r(D) (𝛻𝜙 • 𝑑𝑟⃗ )𝑟⃗Q.12 If a monotonic and continuous function( )y f x= has only one root in theinterval1 2x x x  , then(A)1 2( ) ( ) 0f x f x (B)1 2( ) ( ) 0f x f x =(C)1 2( ) ( ) 0f x f x (D)1 2( ) ( ) 0f x f x− =

Let ϕ(x,y,z)=x2yz+4xz2𝜙(𝑥,𝑦,𝑧)=𝑥2𝑦𝑧+4𝑥𝑧2. The greatest rate of change of this function at the point (1,-2,1) is given bya.39−−√39b.None of thesec.31−−√31d.37−−√

what is a scalar point function

Letf : R^2 -> R: (x,y) -> x^4 + y^2 + 2x^2y -1Answer the following:(a) Graph f(x, 0) against x in two dimensions and graph f(0, y) against y in two dimensions.(b) Draw the countour set f(x, y) = 3(c) Calculate the directional derivative of f at the point (1, 1) in the direction of thevector (−1, 1).(d) At the point (1, 1), find the direction of steepest ascent. That is, find the vector v∗ with length 1 that causes f to increase most rapidly.(e) Find the equation of the tangent plane at the point (1, 1) and comment on the relationship between the tangent plane and the slope of steepest ascent.

A curve in the xy-plane is defined by the vector valued function f, of, tf(t) equals= open angle bracket, x, left bracket, t, right bracket, comma, y, left bracket, t, right bracket, close angle bracket⟨x(t),y(t)⟩, where x, left bracket, t, right bracketx(t) equals= 5, cosine, left bracket, minus, 4, t, right bracket5cos(−4t) and y, left bracket, t, right brackety(t) equals= t, to the power minus 4 , plus, 3t −4 +3 for t, is greater than, 0t>0. What is start fraction, d, squared, y, divided by, d, x, squared, end fraction dx 2 d 2 y​ in terms of tt?

At any x, if first order and second order derivatives of a function f is equal to 0 and greater than 0 respectively, then1 pointx is a local maximum of fx is a local minimum ofx is a stationary pointNone of the above