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Application of Derivatives

Name: Zigya - For The Curious Learner
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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.

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143.

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

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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

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

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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 $left parenthesis 0 comma space straight pi right parenthesis$ .

Here $straight f left parenthesis straight x right parenthesis space equals space sinx$
$therefore space space space space space straight f apostrophe left parenthesis straight x right parenthesis space equals space cos space straight x$
(i) Since for each $straight x space element of space space space open parentheses 0 comma space space straight pi over 2 close parentheses comma space space space space straight f apostrophe left parenthesis straight x right parenthesis space equals space cos space straight x thin space greater than 0$
$therefore space space space space space straight f space space is space strictly space increasing space in space space open parentheses 0 comma space space straight pi over 2 close parentheses$
(ii) Since for each $straight x space element of space space open parentheses straight pi over 2 comma space straight pi close parentheses comma space space straight f apostrophe left parenthesis straight x right parenthesis space equals space cos space straight x thin space less than space 0$
$therefore space space space space space straight f space space is space strictly space decreasing space in space open parentheses straight pi over 2 comma space space straight pi close parentheses.$
(iii) f(x) is strictly increasing in $open parentheses 0 comma space straight pi over 2 close parentheses$ and strictly decreasing in $open parentheses straight pi over 2 comma space straight pi close parentheses$
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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 $left parenthesis negative straight pi comma space straight pi right parenthesis$