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Application of Derivatives

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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)

(1, 2)

The equation of curve is $straight y squared space equals space 4 straight x$
$therefore space space space space space space 2 straight y dy over dx space equals space 4 space space space space space space or space space space space dy over dx space equals space 2 over straight y comma space space space space space which space is space slope space of space tangent space to space the space curve.$
Consider the tangent line y =x+1
Its slope = 1
From given condition,
$2 over straight y space equals space 1 space space space space space or space space space straight y space equals space 2$
When y = 2, 4 = 4x $rightwards double arrow$ x = 1
$therefore space space space space space point space is space left parenthesis 1 comma space 2 right parenthesis therefore space space space space space left parenthesis straight A right parenthesis space is space correct space answer.$

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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

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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.

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

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

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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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