Consider the equation below. (If an answer does not exist, enter DNE.) f(x) = 7 sin(x) + 7 cos(x), 0 ≤ x ≤ 2𝜋

Question

Consider the equation below. (If an answer does not exist, enter DNE.)

f(x) = 7 sin(x) + 7 cos(x),    0 ≤ x ≤ 2𝜋

(a) Find the local minimum and maximum values of f.

local minimum value 

local maximum value

Answer 1

f(x)=7sin(x) + 7cos(x)

f'(x) = 7cos(x) - 7sin(x)

Find critical numbers:

7 cos(x) - 7sin(x) = 0

cos(x) - sin(x) = 0

cos(x) = sin(x)

1 = sin(x)/cos(x) ; cos(x)≠0

1 = tan(x)

x = tan^-1(1)

x = π/4 + kπ, k any integer

When k ε {0,1}, x is in the interval from 0 to 2π, inclusive.

x = π/4, 5π/4

f(π/4) = 7(√2/2) + 7(√2/2) = 7√2 (1st derivative test shows this as relative max)

f(5π/4) = 7(-√2/2) + 7(-√2/2) = -7√2 (relative min)

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