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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)

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

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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, ∞).

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.

Let f (x) = e^x ∴ D_f = R
We are to prove that function f (x) = e^x is strictly increasing. Here interval is not given. So we will prove that function e^x is strictly increasing in its domain.
Now f ' (x) = e^xThree cases arise:
Case I.
$straight x greater than 0$
$Syntax error from line 1 column 415 to line 1 column 518. Unexpected '<mlongdiv '.$

Case II.
x = 0
∴ f ' (x) = e^x = I > 0
Case III.
$straight x less than 0$
$therefore space space space straight f apostrophe left parenthesis straight x right parenthesis space equals space straight e to the power of straight x space equals space 1 over straight e to the power of negative straight x end exponent space equals fraction numerator 1 over denominator straight a space positive space quantitiy space end fraction greater than 0 therefore space space space space space in space all space the space three space cases comma space we space have comma space space space straight f apostrophe left parenthesis straight x right parenthesis space equals space straight e to the power of straight x greater than 0 therefore space space space straight f left parenthesis straight x right parenthesis space equals space straight e to the power of straight x space is space straight a space strictly space increasing space function.$