Short Answer TypeProve that f (x) = ax + b, where a and b are constants and a > 0 is an strictly increasing function for all real values of x. without using the derivative.
Long Answer TypeProve that the function f (x) = sinx is
(i) strictly increasing in 
(ii) strictly decreasing in 
(iii) neither increasing nor decreasing in 
.
Short Answer TypeProve that the function f (x) = cos x is
(i) strictly increasing in 
(ii) strictly decreasing in 
(iii) neither increasing nor decreasing in 
Here f (a) = cos x ⇒ f ' (x) = – sin x
(a) In 
, f '(x) = – sin x > 0
∴ f (x) is strictly increasing in (– 
, 0)
(b) In 
f ' (x) = – sin x < 0
∴ f (x) is strictly decreasing in (0, 
)
(c) Now f (x) is strictly increasing in (– 
, 0) and strictly decreasing in 
∴ f (x) is neither increasing nor decreasing in 
.
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