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Prove that the function f given by f(x) = x – [x] is increasing in (0, 1) ?
we have,
f(x) = x – [x]
∴f'(x) = 1 > 0
∴ f(x) is an increasing function on (0,1)
Find the intervals in which the following functions are increasing or decreasing.
f(x) = 10 – 6x – 2x2
f(x) = x2 + 2x – 5
f(x) = 6 – 9x – x2
f(x) = 2x3 – 12x2 + 18x + 15
f(x) = 5 + 36x + 3x2 – 2x3
f(x) = 8 + 36x + 3x2 – 2x3
f(x) = 5x3 – 15x2 – 120x + 3
f(x) = x3 – 6x2 – 36x + 2
f(x) = 2x3 – 15x2 + 36x + 1
f(x) = 2x3 + 9x2 + 12x + 20
f(x) = 2x3 – 9x2 + 12x – 5
f(x) = 6 + 12x + 3x2 – 2x3
f(x) = 2x3 – 24x + 107
f(x) = – 2x3 – 9x2 – 12x + 1
f(x) = (x – 1) (x – 2)2
f(x) = x3 – 12x2 + 36x + 17
f(x) = 2x3 – 24x + 7
f(x) = x4 – 4x
f(x) = x4 – 4x3 + 4x2 + 15
f(x) = x8 + 6x2
f(x) = x3 – 6x2 + 9x + 15
f(x) = {x (x – 2)}2
f(x) = 3x4 – 4x3 – 12x2 + 5
Determine the values of x for which the function f(x) = x2 – 6x + 9 is increasing or decreasing. Also, find the coordinates of the point on the curve y = x2 – 6x + 9 where the normal is parallel to the liney = x + 5.
Find the intervals in which f(x) = sin x – cos x, where 0 < x < 2π is increasing or decreasing.
Show that f(x) = e2x is increasing on R.
Show that , x ≠ 0 is a decreasing function for all x ≠ 0.
Show that f(x) = loga x, 0 < a < 1 is a decreasing function for all x > 0.
Show that f(x) = sin x is increasing on (0, π/2) and decreasing on (π/2, π) and neither increasing nor decreasing in (0, π).
Show that f(x) = log sin x is increasing on (0, π/2) and decreasing on (π/2, π).
Show that f(x) = x – sin x is increasing for all x ϵ R.
Show that f(x) = x3 – 15x2 + 75x – 50 is an increasing function for all x ϵ R.
Show that f(x) = cos2 x is a decreasing function on (0, π/2).
Show that f(x) = sin x is an increasing function on (–π/2, π/2).
Show that f(x) = cos x is a decreasing function on (0, π), increasing in (–π, 0) and neither increasing nor decreasing in (–π, π).
Show that f(x) = tan x is an increasing function on (–π/2, π/2).
Show that f(x) = tan–1 (sin x + cos x) is a decreasing function on the interval (π/4, π /2).
Show that the function is decreasing on
Show that the function f(x) = cot–1 (sin x + cos x) is decreasing on (0, π/4) and increasing on (π/4, π/2).
Show that f(x) = (x – 1) ex + 1 is an increasing function for all x > 0.
Show that the function x2 – x + 1 is neither increasing nor decreasing on (0, 1).
Show that f(x) = x9 + 4x7 + 11 is an increasing function for all x ϵ R.
Prove that the function f(x) = x3 – 6x2 + 12x – 18 is increasing on R.
State when a function f(x) is said to be increasing on an interval [a, b]. Test whether the function f(x) = x2 – 6x + 3 is increasing on the interval [4, 6].
Show that f(x) = sin x – cos x is an increasing function on (–π /4, π /4)?
Show that f(x) = tan–1 x – x is a decreasing function on R ?
Determine whether f(x) = x/2 + sin x is increasing or decreasing on (–π /3, π/3) ?
Find the interval in which is increasing or decreasing ?
Find the intervals in which f(x) = (x + 2)e–x is increasing or decreasing ?
Show that the function f given by f(x) = 10x is increasing for all x ?
Prove that the following function is increasing on r?
i. f(x) = 3x5 + 40x3 + 240x
ii. f(x) = 4x3 – 18x2 + 27x – 27
Prove that the function f given by f(x) = log cos x is strictly increasing on (–π/2, 0) and strictly decreasing on (0, π/2) ?
Prove that the function f given by f(x) = x3 – 3x2 + 4x is strictly increasing on R ?
33 Prove that the function f(x) = cos x is :
i. strictly decreasing on (0, π)
ii. strictly increasing in (π, 2π)
iii. neither increasing nor decreasing in (0, 2 π)
Show that f(x) = – x sin x is an increasing function on (0, π/2) ?
Find the value(s) of a for which f(x) = – ax is an increasing function on R ?
Find the values of b for which the function f(x) = sin x – bx + c is a decreasing function on R ?
Show that f(x) = x + cos x – a is an increasing function on R for all values of a ?
Let F defined on [0, 1] be twice differentiable such that | f”(x) ≤ 1 for all x ϵ [0, 1]. If f(0) = f(1), then show that |f’(x) | < 1 for all x ϵ [0, 1] ?
Find the intervals in which f(x) is increasing or decreasing :
i. f(x) = x |x|, x ϵ R
ii. f(x) = sin x + |sin x|, 0 < x ≤ 2 π
iii. f(x) = sin x (1 + cos x), 0 < x < π/2