Limits and Derivatives

Suppose that and if , what are possible values of a and b? - Wired Faculty

Question

Suppose that

$space space space space space space space space space space space straight f left parenthesis straight x right parenthesis space open curly brackets table row cell table row cell table row cell straight a plus bx comma space straight x space less than 1 end cell end table end cell end table end cell row cell table row cell table row cell 4 comma space straight x space equals space 1 end cell row cell straight b minus ax comma space straight x space greater than 1 end cell end table end cell end table end cell end table close$

and if $<pre>uncaught exception: mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56 #0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>$ , what are possible values of a and b?

Solution

L.H.L. = $space space space space space space space space space limit as straight x rightwards arrow 1 to the power of minus of straight f left parenthesis straight x right parenthesis space equals limit as straight x rightwards arrow 1 to the power of minus of left parenthesis straight a plus bx right parenthesis$ [∵ here x <1]
Put x = 1-4, h>0 and h $rightwards arrow$ 0
∴ L.H.L = $space space space space space space space space space space space stack space lim with h rightwards arrow 0 below left square bracket a plus b left parenthesis 1 minus h right parenthesis right square bracket space equals space a plus b left parenthesis 1 minus 0 right parenthesis equals a plus b$

R.H.L. = $space space space space limit as straight x rightwards arrow 1 to the power of plus of straight f left parenthesis straight x right parenthesis space equals space limit as straight x rightwards arrow 1 to the power of plus of left parenthesis straight b minus ax right parenthesis$ [∵ here x > 1]
Put x = 1-h, h>0 and h $rightwards arrow$ 0
∴ R.H.L = $space space space space space space space limit as straight h rightwards arrow 0 of left square bracket straight b minus straight a left parenthesis 1 plus straight h right parenthesis right square bracket equals straight b minus straight a left parenthesis 1 plus 0 right parenthesis equals straight b minus straight a$
Also, f(1) = 4
Now, $<pre>uncaught exception: mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56 #0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>$
$rightwards double arrow$
$space space space space space space limit as straight x rightwards arrow 1 to the power of minus of straight f left parenthesis straight x right parenthesis space equals straight f left parenthesis 1 right parenthesis space and space limit as straight x rightwards arrow 1 to the power of plus of straight f left parenthesis straight x right parenthesis space equals space straight f left parenthesis 1 right parenthesis$
$<pre>uncaught exception: mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56 #0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>$ a + b = 4 adn b - a = 4
Adding these equations, we get 2b = 8 $rightwards double arrow$ b = 4
∴ a = 4 - b = 4 - 4 = 0
Hence, a = 0, b = 4

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