Given the function below f(x, y, z) = 2 cos (r) + xy - xz - y² - e²% +2% (a) Show that the origin (0, 0, 0) is a stationary point (show the gradient is 0).

2. For the function z = f(x, y) is known f (2, – 1) = -2, f-(2, -1)=-1, f,(2, –1) = 2. Find the following (a) Gradient of f at the point (2, –1) :grad f(2, -1) (b) Directional derivative of f at the point (2, – 1) in the direction of the vector –27+37: fa(2, –1 ). (c) Estimate f(1.99, –1.02) = (d) Differential of f at the point (2, -1)

Calculate the gradient of the function f(T, y) = cos (4x - 4y) O a. -8x sin (4x2 4y)i + 4 sin (4r2 - 4y)j O b. 8r sin (4x² – 4y)i – 4 sin (4x² – 4y)j O c.- sin (4x2 4y)i + sin (4x² 4y)j O d. sin (4x - 4y)i sin (4x² – 4y)j

Read the following proof and identify proof steps. 1. Here we will prove that the sum of the first n odd integers is n2, That is, Vn > 0 E,(2n – 1) = n? i=1 2. We will use mathematical induction on E (2n – 1) = n² i=1 3. For n=1, E (2 ·1– 1) = 2 · 1 -1=1= 12 'i=1 Base case Induction step 4. Assume that E(2k – 1) = k² Theorem statement 5. E" (2(k+1) – 1) = (E(2k – 1)) + (2(k+1) – 1) = k² + (2(k+1) – 1) = k² + 2k +1= (k+ 1)² Induction hypothesis i31 6. Therefore, Vn > 0 E,(2n – 1) = n?