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Here f (x) = x 3 – 3x 2 + 4x D f = R f '(x) = 3x 2 – 6x + 4 = 3 (x 2 – 2x + 1) + 1 = 3 (x – 1) 2 + 1 > 0 ∀ x ∊ R ∴ f is strictly increasing on R.

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Application of Derivatives

Short Answer Type

141.

Show that the function given by f(x) = e^2x is strictly increasing on R.

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142.

Prove 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.

88 Views

143.

Prove that the logarithmic function is increasing wherever it is defined.

86 Views

144.

Show that the function f given by f (x) = 10^x is increasing for all x.

85 Views

145.

Prove that the function f(x) = x³ – 3x² + 3x – 100 is increasing on R.

88 Views

146.

Prove that the function f(x) = x³ – 3x² + 3x – 100 is increasing on R.

90 Views

147. Prove that the function f (x) = 4x³ – 6x² + 3x + 12 is increasing on R

94 Views

148. Show that the function f given by f(x) = x³ – 3x² + 4x , x ∊ R is strictly increasing on R.

Here f (x) = x³– 3x² + 4x
D_f = R
f '(x) = 3x² – 6x + 4 = 3 (x² – 2x + 1) + 1 = 3 (x – 1)² + 1 > 0 ∀ x ∊ R
∴ f is strictly increasing on R.

95 Views

Long Answer Type

149.

Prove that the function f (x) = sinx is
(i) strictly increasing in $open parentheses 0 comma space straight pi over 2 close parentheses$
(ii) strictly decreasing in $open parentheses straight pi over 2 comma space straight pi close parentheses$
(iii) neither increasing nor decreasing in $left parenthesis 0 comma space straight pi right parenthesis$ .

88 Views

Short Answer Type

150.

Prove that the function f (x) = cos x is
(i) strictly increasing in $left parenthesis negative straight pi comma space 0 right parenthesis$
(ii) strictly decreasing in $left parenthesis 0 comma space straight pi right parenthesis$
(iii) neither increasing nor decreasing in $left parenthesis negative straight pi comma space straight pi right parenthesis$