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Let x 1 , x 2 ∊ R and let x 1 < x 2 Now x 1 , < x 2 ⇒ 2x 1 < 2x 2 ⇒ 2x 1 + 3 < 2x 2 + 3 ⇒ f (x 1 ) < f (x 2 ) ⇒ f is an increasing function on R.

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Application of Derivatives

Multiple Choice Questions

131.

The slope of the normal to the curve y = 2 x² + 3 sin x at x = 0 is

3
$1 third$
-3
-3

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132.

The line y = x + 1 is a tangent to the curve y ² = 4x at the point

(1, 2)
(2, 3)
(1, -2)
(1, -2)

75 Views

133. The normal at the point (1, 1) on the curve 2y + x² = 3 is

x + y = 0
x – y = 0
x + y + 1 = 0
x + y + 1 = 0

114 Views

134. The normal to the curve x² = 4 y passing (1, 2) is

x + y = 3
x – y = 3
x + y = 1
x + y = 1

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Short Answer Type

135. Show that the function f (x) = 2 x + 3 is a strictly increasing function on R.

Let x₁, x₂ ∊ R and let x₁ < x₂Now x₁, < x₂⇒ 2x₁ < 2x₂⇒ 2x₁ + 3 < 2x₂+ 3
⇒ f (x₁) < f (x₂)
⇒ f is an increasing function on R.

85 Views

136. Without using the derivative show that the function f (a) = 7x – 3 is a strictly increasing function on R.

85 Views

137. Show that the function f (x) = x² is an increasing function in (0, ∞).

81 Views

138.

Show that the function f(x) = x² is a decreasing function in (– ∞ 0).

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139.

Construct an example of a functions which is strictly increasing but whose derivative vanishes at a point in the domain of definition of the function.

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140.

Prove that the exponential function e^x is strictly increasing on R.

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