Determine the int

Given: f(x) = x⁴ - 8x³ + 22x² - 24x + 21

Finding f ^‘(x)

f ^‘(x) = 4x³ - 24x² + 44x - 24

formula:

f ^‘(x) = 4(x³ - 6x² + 11x - 6)

f ^‘(x) = 4(x - 1) (x - 2) (x - 3)

for finding the maxima or minima f ^‘(x) = 0 at that point

f ^‘(x) = 0 at x = 1, x = 2 and x = 3

the intervals are ( - ∞, 1) (1, 2) (2, 3) (3, ∞)

since f ^‘(x) > 0 in (1, 2) and (3, ∞)

therefore f(x) is strictly increasing in (1, 2) and (3, ∞) and strictly decreasing in ( - ∞, 1) and (2, 3).

OR

Given: f(x) = sec x + log cos²x

Finding f ^‘(x)

f ^‘(x) = sec x tan x - 2tan x

Formula:

f ^‘(x) = tan x (sec x - 2)

for finding the maxima or minima f ^‘(x) = 0 at that point

f ^‘(x) = 0 at tan x = 0 or sec x = 2

therefore, solution is x = π or

now finding f” (x) to check whether the point is maxima or minima

f” (x) = sec x tan²x + (sec x - 2) sec²x

Formula:

substituting the points

which is positive hence it is minimum

which is negative hence it is maximum

which is positive hence it is minimum

Therefore, we can conclude that

Maximum value

Minimum value