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Hi there.
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In the question it is given that f of x is equal to natural logarithm of 3x minus 4x cube, we have to find the numbers x in the domain of f of x, for which the tangent line is horizontal.
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Let's see how we'll do this.
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First we'll find the domain of f of x.
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We know that the function f of x is natural logarithm of 3x minus 4x cube.
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So this is defined.
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The logarithm is defined only if this particular thing is greater than 0.
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That means the 3x minus 4x cube should be greater than 0.
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So we'll have 3x minus 4x cube is greater than 0.
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And this implies that x multiplied by 3 minus 4x square is greater than 0.
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So here there will be two cases.
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Case 1 is that x is greater than 0 and 3 minus 4x squared.
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4x square is greater than 0.
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This implies that 4x square is less than 3 and this implies that x square is less than 3 divided by 4.
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So from here we'll be getting that x is less than root 3 divided by 2 or x is greater than negative root 3 divided by 2.
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So from this, this, this we'll be getting that.
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The first interval satisfying this condition that is x multiplied by 3 minus 4x square greater than 0 is the interval is open interval 0 comma root 3 divided by 2...