document.write( "Question 1184172: Can we apply role's theorem in the interval [-2, 2] for following functions:
\n" ); document.write( "f(x) = 4.x^4 and g(x) = πx? \n" ); document.write( "

Algebra.Com's Answer #814711 by MathLover1(20850) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Rolle’s Theorem - Let $\"f%28x%29\"$ be continuous on the closed interval [ $\"a\"$ , $\"b\"$ ] and differentiable on the interval [ $\"a\"$ , $\"b\"$ ]. \r
\n" ); document.write( "\n" ); document.write( "If $\"f%28+a%29+=f%28b%29\"$ then there is at least one number $\"c\"$ in [ $\"a\"$ , $\"b\"$ ] such that $\"f\"$ ' $\"%28c%29=0\"$ . \r
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\n" ); document.write( "\n" ); document.write( " $\"f%28x%29+=+4x%5E4+\"$ in the interval [ $\"-2\"$ , $\"+2\"$ ] \r
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\n" ); document.write( "\n" ); document.write( "check:
\n" ); document.write( "if continuous on [ $\"-2\"$ , $\"+2\"$ ]
\n" ); document.write( "if differentiable on [ $\"-2\"$ , $\"+2\"$ ]
\n" ); document.write( "if $\"f%28-2%29+=f%282%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "The function $\"f%28x%29+=4x%5E4\"$ is continuous and differentiable on [ $\"-2\"$ , $\"+2\"$ ].\r
\n" ); document.write( "\n" ); document.write( " $\"f%28-2%29+=+4%28-2%29%5E4=64\"$
\n" ); document.write( " $\"f%282%29+=+4%282%29%5E4=64\"$ => $\"f%28-2%29+=f%282%29\"$ \r
\n" ); document.write( "\n" ); document.write( "=> role's theorem can be applied\r
\n" ); document.write( "\n" ); document.write( " $\"f\"$ ' $\"%28x%29+=+16x%5E3\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"f\"$ ' $\"+%28c%29=0\"$ gives $\"16x%5E3=0\"$ => $\"c=0\"$ \r
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\n" ); document.write( "\n" ); document.write( "and\r
\n" ); document.write( "\n" ); document.write( " $\"g%28x%29+=+pi%2Ax\"$ \r
\n" ); document.write( "\n" ); document.write( "is continuous and differentiable on [ $\"-2\"$ , $\"2\"$ ]
\n" ); document.write( " $\"g%28-2%29+=+pi%2A%28-2%29=-2pi\"$
\n" ); document.write( " $\"g%282%29+=+pi%2A%282%29=2pi\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"g\"$ ' $\"%28x%29+=pi\"$ \r
\n" ); document.write( "\n" ); document.write( "=> since $\"g%28-2%29+%3C%3Eg%282%29\"$ , role's theorem cannot be applied\r
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