Long Answer TypeFind intervals in which the function given by 
is (a) strictly increasing (b) strictly decreasing.
Short Answer TypeSeparate 
into sub-intervals in which the function f (x) = sin 3x is increasing or decreasing.
Long Answer TypeFind the intervals in which the following function is increasing or decreasing:
f (x) = sinx – cosx, 0 < x < 2
.
Short Answer TypeFind the intervals in which the following function is increasing or decreasing
f (x) = (x + 2) e–x
f (x) = (x + 2) e–x
∴ f ' (x) = (x + 2) · e–x (– 1) + e–x · 1 = e–x (– x – 2 + 1)
∴ f ' (x) = – (x + 1) e–x
For f (x) to be increasing
f ' (x) > 0 ⇒ –(x + 1) e–x > 0
⇒ (x + 1) e–x < 0 ⇒ x + 1< 0 [∵ e–x > 0]
⇒ x < – 1
∴ f (x) is increasing in ( – ∞ , – 1)
For f (x) to be decreasing,
f ' (x) < 0 ⇒ – (x + 1) e–x < 0
⇒ (x + 1) e–x > 0 ⇒ x + 1 > 0 [ ∵ c–x > 0]
⇒ x > – 1
∴ f (x) is decreasing in (– 1, ∞)
Long Answer Type
is increasing or decreasing.Find the intervals in which the function (x + 1)3 (x – 1)3 is strictly increasing or decreasing.
Short Answer TypeLet f be a function defined on [a, b] such that f ' (x) > 0, for all x ∊ (a, b). Then prove that f is strictly increasing function of (a, b).
On which of the following intervals is the function f given by f (x) = x100 + sin x – 1 strictly decreasing?
(A) (0, 1) (B) 
(C) 
(D) None of these
is increasing or decreasing.
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