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bekas [8.4K]
3 years ago
10

If f(x)=ln(sin(2x)), f''(π/4) is equal to​

Mathematics
1 answer:
Licemer1 [7]3 years ago
4 0

Use the chain rule to compute the second derivative:

f(x)=\ln(\sin(2x))

The first derivative is

f'(x)=(\ln(\sin(2x)))'=\dfrac{(\sin(2x))'}{\sin(2x)}=\dfrac{\cos(2x)(2x)'}{\sin(2x)}=\dfrac{2\cos(2x)}{\sin(2x)}

f'(x)=2\cot(2x)

Then the second derivative is

f''(x)=(2\cot(2x))'=-2\csc^2(2x)(2x)'

f''(x)=-4\csc^2(2x)

Then plug in π/4 for <em>x</em> :

f''\left(\dfrac\pi4\right)=-4\csc^2\left(\dfrac{2\pi}4\right)=-4

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\\  \sf \: 3 +  \sqrt{5}  =  \frac{a}{b}   \\  \\   \qquad \: \tiny \sf{(where \:  \: a \:  \: and \:  \: b \:  \: are \:  \: integers \:  \: and \:  \: b \:  \neq \: 0)} \\

\\  \sf \:  \sqrt{5}  =  \frac{a}{b}  - 3 =  \frac{a - 3b}{b}  \\

Since, a, b and 3 are integers. So,

\\ \sf \:  \frac{p - 3b}{b}  \\  \\  \qquad \tiny \sf{ \: (is \:  \: a \:  \: rational \:  \: number \:) } \\

Here, it contradicts that √5 is an irrational number.

because of the wrong assumption that 3 + √5 is a rational number.

\\

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