Show that a differentiable function f decreases most rapidly at x in the direction opposite the gradient vector, that is, in the direction of −∇f(x). let θ be the angle between ∇f(x) and unit vector u. then du f = |∇f| . since the minimum value of is occurring, for 0 ≤ θ < 2π, when θ = , the minimum value of du f is −|∇f|, occurring when the direction of u is the direction of ∇f (assuming ∇f is not zero).

the correct answer is b) histograms can be used to exhibit the shape of distributions. this is because histograms count the frequency of each possible answer, so one is able to see which results are more common and which are less common.

answer:

it is general formula that is used to write value to its excet position

a11 a12 a13

a21 a22 a23

it is the example of order 2 by 3 matrix in this

a11 means value occur in 1 st row 1st column

a12 means value occurs in 1st row 2nd column

a13 means value occur in 1st row and 3rd column

a21 means value occurs in 2nd row and 1st column

a22 means value occurs in 2nd row and in 2nd column

a23 means value occurs in 2nd row and in 3rd column

step-by-step explanation:

jbjuiuhuhuuy

step-by-step explanation:

what does 'mandy have ben 780 bottle caps. ' mean?