Find the absolute maximum and the absolute minimum values of function f(x)= 3x² - 6x + 4 in [-1, 2].

Ans: Absolute max. value = 13 at x = -1; Absolute min. value = 1 at x = 1

Determine the interval in which the function f(x) = (1/2) x² - x is increasing or decreasing.

Ans: Decreasing on (- ∞, 1) and Increasing on (1, ∞)

For any curve y = f(x) State what do f ' (x) > 0 and f ' (x) < 0 represent.

Ans: Increasing function and decreasing function

Test whether the function f(x) = 2x² - 4x + 3 is increasing or decreasing on the interval [1, 4].

Ans: Increasing

Find the minimum value of the function 2x² + 4x + 7.

Ans: Min. value = 5 at x = -1.

Examine nature of the function y = x - (1/x) for increasing or decreasing

Ans: Increasing for all x ∈ R except x = 0

Show that the function f(x) = 2x³ - 24x + 15 is increasing at x = 3 and decreasing at x = 3/2.

Ans: Increasing at x = 3, Decreasing at x = 3/2

Examine whether the function f(x) = 15x²- 14x + 1 is increasing or decreasing at x = 2/5 and x = 5/2.

Ans: Decreasing at x = 2/5, Increasing at x = 5/2

Find the intervals in which the function f(x) = 5x³ – 135x + 22 is increasing or decreasing.

Ans: Increasing on (-∞, -3) U (3, ∞) and Decreasing on (-3, 3)

Find the intervals in which the function f(x) = 3x² - 6x + 5 is increasing and decreasing.

Ans: f(x) is decreasing on (- ∞, 1) and increasing on (1, ∞)

Show that f(x) = x - (1/x), is increasing for all x ∈ R except at x = 0.

If f(x) = 2x³ - 6x2 + 5, determine where does the graph of the function concave upward.

Ans: (1, ∞)

Show that the function y = x3 - 3x² + 6x + 3 has neither maximum nor minimum values.

Find the intervals in which f(x) = x² - 2x + 10 is increasing or decreasing.

Ans: Decreasing on (- ∞, 1) and Increasing on (1, ∞)

Find the minimum of f(x) = 3x² - 6x +4.

Ans: 1