{"id":20185,"date":"2024-04-15T05:49:30","date_gmt":"2024-04-15T05:49:30","guid":{"rendered":"https:\/\/exam.pscnotes.com\/mcq\/?p=20185"},"modified":"2024-04-15T05:49:30","modified_gmt":"2024-04-15T05:49:30","slug":"directional-derivative-of-phi-2xz-y2-at-the-point-1-3-2-becomes-maximum-in-the-direction-of-a-rm4hat-i-2rmhat-j-3rmhat-k-b-rm4hat-i-6","status":"publish","type":"post","link":"https:\/\/exam.pscnotes.com\/mcq\/directional-derivative-of-phi-2xz-y2-at-the-point-1-3-2-becomes-maximum-in-the-direction-of-a-rm4hat-i-2rmhat-j-3rmhat-k-b-rm4hat-i-6\/","title":{"rendered":"Directional derivative of \\[\\phi \\] = 2xz &#8211; y2 at the point (1, 3, 2) becomes maximum in the direction of: A. \\[{\\rm{4\\hat i}} + 2{\\rm{\\hat j}} &#8211; 3{\\rm{\\hat k}}\\] B. \\[{\\rm{4\\hat i}} &#8211; 6{\\rm{\\hat j}} + 2{\\rm{\\hat k}}\\] C. \\[{\\rm{2\\hat i}} &#8211; 6{\\rm{\\hat j}} + 2{\\rm{\\hat k}}\\] D. \\[{\\rm{4\\hat i}} &#8211; 6{\\rm{\\hat j}} &#8211; 2{\\rm{\\hat k}}\\]"},"content":{"rendered":"<p>\r\n    <!-- Check if it's an AMP page -->\r\n            <!-- Non-AMP version -->\r\n        <div class=\"mcq-container\" data-quiz-id=\"quizState_6a9ba5fa5184a\">\r\n                                            <div class=\"option\" data-option-key=\"option1\" data-is-correct=\"false\">\r\n                    &#8221;[{\rm{4hat                <\/div>\r\n                                                                                                                            \r\n            <!-- Feedback messages for non-AMP -->\r\n            <div class=\"feedback\" data-feedback=\"wrong\">Answer is Right!<\/div>\r\n            <div class=\"feedback\" data-feedback=\"right\">Answer is Wrong!<\/div>\r\n        <\/div>\r\n\r\n        <script>\r\n        document.addEventListener('DOMContentLoaded', function () {\r\n            var containers = document.querySelectorAll('.mcq-container');\r\n\r\n            containers.forEach(function(container) {\r\n                var options = container.querySelectorAll('.option');\r\n                var feedbackSelect = container.querySelector('[data-feedback=\"select\"]');\r\n                var feedbackWrong = container.querySelector('[data-feedback=\"wrong\"]');\r\n                var feedbackRight = container.querySelector('[data-feedback=\"right\"]');\r\n\r\n                options.forEach(function(option) {\r\n                    option.addEventListener('click', function() {\r\n                        var selectedOption = option.getAttribute('data-option-key');\r\n                        var isCorrect = option.getAttribute('data-is-correct') === 'true';\r\n\r\n                        \/\/ Remove previous selections\r\n                        options.forEach(function(opt) {\r\n                            opt.classList.remove('correct', 'incorrect');\r\n                        });\r\n\r\n                        \/\/ Add the correct\/incorrect class\r\n                        if (isCorrect) {\r\n                            option.classList.add('correct');\r\n                            feedbackRight.hidden = false;\r\n                            feedbackWrong.hidden = true;\r\n                        } else {\r\n                            option.classList.add('incorrect');\r\n                            feedbackRight.hidden = true;\r\n                            feedbackWrong.hidden = false;\r\n                        }\r\n\r\n                        \/\/ Hide select feedback\r\n                        feedbackSelect.hidden = true;\r\n                    });\r\n                });\r\n            });\r\n        });\r\n        <\/script>\r\n    \r\n    &#8221; option2=&#8221;\\[{\\rm{4\\hat i}} &#8211; 6{\\rm{\\hat j}} + 2{\\rm{\\hat k}}\\]&#8221; option3=&#8221;\\[{\\rm{2\\hat i}} &#8211; 6{\\rm{\\hat j}} + 2{\\rm{\\hat k}}\\]&#8221; option4=&#8221;\\[{\\rm{4\\hat i}} &#8211; 6{\\rm{\\hat j}} &#8211; 2{\\rm{\\hat k}}\\]&#8221; correct=&#8221;option3&#8243;]<!--more--><\/p>\n<p>The correct answer is $\\boxed{\\text{(B)}}$.<\/p>\n<p>The directional derivative of a function $\\phi$ at a point $P$ in the direction of a <div class=\"youtube-subscribe-container\">\r\n        <a href=\"https:\/\/www.youtube.com\/channel\/UCNHT8lW-JmLC68rjBfZhdkg?sub_confirmation=1\" target=\"_blank\" class=\"youtube-subscribe-button\">\r\n            <span class=\"youtube-icon\">\r\n                <svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" viewBox=\"0 0 576 512\">\r\n                    <path d=\"M549.7 124.1c-6.3-23.7-24.8-42.3-48.3-48.6C458.8 64 288 64 288 64S117.2 64 74.6 75.5c-23.5 6.3-42 24.9-48.3 48.6-11.4 42.9-11.4 132.3-11.4 132.3s0 89.4 11.4 132.3c6.3 23.7 24.8 41.5 48.3 47.8C117.2 448 288 448 288 448s170.8 0 213.4-11.5c23.5-6.3 42-24.2 48.3-47.8 11.4-42.9 11.4-132.3 11.4-132.3s0-89.4-11.4-132.3zm-317.5 213.5V175.2l142.7 81.2-142.7 81.2z\"\/>\r\n                <\/svg>\r\n            <\/span>\r\n            Subscribe on YouTube\r\n       <div class=\"telegram-channel-container\">\r\n        <a href=\"https:\/\/t.me\/pscnotes2025\" target=\"_blank\" class=\"telegram-channel-button\">\r\n            <span class=\"telegram-icon\">\r\n                <svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" viewBox=\"0 0 496 512\">\r\n                    <path fill=\"white\" d=\"M248,8C111,8,0,119,0,256s111,248,248,248s248-111,248-248S385,8,248,8z M362,177L320,367c-3,14-10,18-20,14l-56-41l-27,26 c-3,3-5,5-10,5l4-63L323,196c5-5-1-7-8-3l-98,62l-42-13c-9-3-10-9,2-14l162-63C351,160,365,164,362,177z\"\/>\r\n                <\/svg>\r\n            <\/span>\r\n            Join Our Telegram Channel\r\n        <\/a>\r\n    <\/div>  <\/a>\r\n    <\/div> unit vector $\\hat{u}$ is given by:<\/p>\n<p>$$D_{\\hat{u}} \\phi(P) = \\nabla \\phi(P) \\cdot \\hat{u}$$<\/p>\n<p>where $\\nabla \\phi(P)$ is the gradient of $\\phi$ at $P$.<\/p>\n<p>In this case, we have:<\/p>\n<p>$$\\phi(x, y, z) = 2xz &#8211; y^2$$<\/p>\n<p>$$\\nabla \\phi(x, y, z) = (2x + 2z) \\hat{i} + (2y) \\hat{j} &#8211; 2y \\hat{k}$$<\/p>\n<p>Substituting $x = 1$, $y = 3$, and $z = 2$, we get:<\/p>\n<p>$$\\nabla \\phi(1, 3, 2) = (4 + 4) \\hat{i} + (6) \\hat{j} &#8211; (4) \\hat{k} = 4 \\hat{i} &#8211; 6 \\hat{j} + 2 \\hat{k}$$<\/p>\n<p>The directional derivative of $\\phi$ at the point $(1, 3, 2)$ in the direction of $\\hat{u}$ is then:<\/p>\n<p>$$D_{\\hat{u}} \\phi(1, 3, 2) = (4 \\hat{i} &#8211; 6 \\hat{j} + 2 \\hat{k}) \\cdot \\hat{u}$$<\/p>\n<p>The maximum value of the directional derivative occurs when $\\hat{u}$ is in the same direction as $\\nabla \\phi(1, 3, 2)$. Therefore, the maximum value of the directional derivative is:<\/p>\n<p>$$\\max_{\\hat{u}} D_{\\hat{u}} \\phi(1, 3, 2) = (4 \\hat{i} &#8211; 6 \\hat{j} + 2 \\hat{k}) \\cdot (4 \\hat{i} &#8211; 6 \\hat{j} + 2 \\hat{k}) = 24$$<\/p>\n<p>Therefore, the directional derivative of $\\phi$ at the point $(1, 3, 2)$ becomes maximum in the direction of $\\boxed{\\text{(B)}}$.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>&#8221; option2=&#8221;\\[{\\rm{4\\hat i}} &#8211; 6{\\rm{\\hat j}} + 2{\\rm{\\hat k}}\\]&#8221; option3=&#8221;\\[{\\rm{2\\hat i}} &#8211; 6{\\rm{\\hat j}} + 2{\\rm{\\hat k}}\\]&#8221; option4=&#8221;\\[{\\rm{4\\hat Join Our Telegram Channel Subscribe on YouTube i}} &#8211; 6{\\rm{\\hat j}} &#8211; 2{\\rm{\\hat k}}\\]&#8221; correct=&#8221;option3&#8243;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[690],"tags":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v22.2 (Yoast SEO v23.3) - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Directional derivative of \\[\\phi \\] = 2xz - y2 at the point (1, 3, 2) becomes maximum in the direction of: A. \\[{\\rm{4\\hat i}} + 2{\\rm{\\hat j}} - 3{\\rm{\\hat k}}\\] B. \\[{\\rm{4\\hat i}} - 6{\\rm{\\hat j}} + 2{\\rm{\\hat k}}\\] C. \\[{\\rm{2\\hat i}} - 6{\\rm{\\hat j}} + 2{\\rm{\\hat k}}\\] D. \\[{\\rm{4\\hat i}} - 6{\\rm{\\hat j}} - 2{\\rm{\\hat k}}\\]<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/exam.pscnotes.com\/mcq\/directional-derivative-of-phi-2xz-y2-at-the-point-1-3-2-becomes-maximum-in-the-direction-of-a-rm4hat-i-2rmhat-j-3rmhat-k-b-rm4hat-i-6\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Directional derivative of \\[\\phi \\] = 2xz - y2 at the point (1, 3, 2) becomes maximum in the direction of: A. \\[{\\rm{4\\hat i}} + 2{\\rm{\\hat j}} - 3{\\rm{\\hat k}}\\] B. \\[{\\rm{4\\hat i}} - 6{\\rm{\\hat j}} + 2{\\rm{\\hat k}}\\] C. \\[{\\rm{2\\hat i}} - 6{\\rm{\\hat j}} + 2{\\rm{\\hat k}}\\] D. \\[{\\rm{4\\hat i}} - 6{\\rm{\\hat j}} - 2{\\rm{\\hat k}}\\]\" \/>\n<meta property=\"og:description\" content=\"&#8221; 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