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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

-3

The equation of curve is
y = 2x² + 3 sin x
$therefore space space space space space space space space space dy over dx space equals space 4 space straight x plus 3 space cosx At space space space straight x space equals space 0 comma space space space dy over dx space equals space 4 left parenthesis 0 right parenthesis space plus space 3 space cos space 0 space equals space 0 plus 3 space equals space 3 comma space which space is space slope space of space tangent space to space the space curve. therefore space space space space space space space slope space of space normal space to space the space curve space equals space minus 1 third therefore space space space space space space space left parenthesis straight D right parenthesis space is space correct space answer.$

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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)

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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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