critical cos(5x),0<= x<= 2pi

Solution

critical points

cos (5 x), 0 \leq x \leq 2 π

x = 0, x = \frac{π}{5}, x = \frac{2 π}{5}, x = \frac{3 π}{5}, x = \frac{4 π}{5}, x = π, x = \frac{6 π}{5}, x = \frac{7 π}{5}, x = \frac{8 π}{5}, x = \frac{9 π}{5}, x = 2 π

Solution steps

cos (5 x), 0 \leq x \leq 2 π

Find where

f^{'} (x)

is equal to zero or undefined

x = \frac{π}{5}, x = \frac{2 π}{5}, x = \frac{3 π}{5}, x = \frac{4 π}{5}, x = π, x = \frac{6 π}{5}, x = \frac{7 π}{5}, x = \frac{8 π}{5}, x = \frac{9 π}{5}

Identify critical points not in

f (x)

domain

Domain of

cos (5 x), 0 \leq x \leq 2 π : 0 \leq x \leq 2 π

All critical points are in domain

x = \frac{π}{5}, x = \frac{2 π}{5}, x = \frac{3 π}{5}, x = \frac{4 π}{5}, x = π, x = \frac{6 π}{5}, x = \frac{7 π}{5}, x = \frac{8 π}{5}, x = \frac{9 π}{5}

Evaluate interval endpoint

x = 0 :

critical point

Evaluate interval endpoint

x = 2 π :

critical point

x = 0, x = \frac{π}{5}, x = \frac{2 π}{5}, x = \frac{3 π}{5}, x = \frac{4 π}{5}, x = π, x = \frac{6 π}{5}, x = \frac{7 π}{5}, x = \frac{8 π}{5}, x = \frac{9 π}{5}, x = 2 π

2 x e^{x^{2}}

x^{4} - 18 x^{2} + 1

f (x) = I n (\frac{x ^{3} + 9 x ^{2} - 10 x}{4 x ^{3} - 4})

f (x) = x^{- \frac{1}{3}} (x - 16)

2 t^{3} - 24 t^{2} + 72 t + 30