document.write( "Question 1192279: The gradient/slope of a function of f(x) = (2x-b)^a is f'(x) = 24x^2-24x+6, find the constant a and b. I'm a confused on how to start on this. \n" ); document.write( "

Algebra.Com's Answer #824188 by ikleyn(52786) $\"\"$ $\"About$

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\n" ); document.write( "The gradient/slope of a function of f(x) = (2x-b)^a is f'(x) = 24x^2-24x+6,
\n" ); document.write( "find the constant a and b. I'm a confused on how to start on this.
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document.write( "The wording of the problem means that the derivative of the function f(x) = (2x-b)^a\r\n" );
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document.write( "is the polynomial of degree 2,  f'(x) = 24x^2 - 24x + 6.\r\n" );
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document.write( "It means that f(x) is a polynomial of degree 3;  hence, a= 3.\r\n" );
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document.write( "Next, you can make FOIL  of  (2x-b)^3  to get  (2x-b)^3 = 8x^3 - 3*(2x^2)*b + . . . = 8x^2 - 12x^2*b + . . . \r\n" );
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document.write( "Then, looking at its term of degree 2,  -3*(2x^2)*b = -3*(4x^2)*b = -12x^2*b,  and comparing it \r\n" );
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document.write( "with its derivative -24x, you conclude that b= 1.\r\n" );
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document.write( "So,  the constants \"a\" and \"b\" are  a= 3,  b= 1.    \r\n" );
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document.write( "Thus  f(x) = (2x-1)^3 = 8x^3 - 3*(2x)^2*1 + 3*(2x)*1^2 - 1 = 8x^3 - 12x^2 + 6x - 1.\r\n" );
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document.write( "ANSWER.  a= 3, b= 1.\r\n" );
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