To find the relative extrema of the function We use the power rule:<\/p>\n\n\n\n f(x) = 3x – 36x^(1\/3) We solve for critical points by setting f\u2032(x) = 0:<\/p>\n\n\n\n 3 – 12x^(-2\/3) = 0 x^(-2\/3) = 1\/4 x = \u00b1(4)^(3\/2) = \u00b18<\/p>\n\n\n\n So we get two critical points: x = 8 and x = -8<\/p>\n\n\n\n Second derivative:<\/strong><\/p>\n\n\n\n f\u2032(x) = 3 – 12x^(-2\/3) f\u2033(x) = d\/dx[-12x^(-2\/3)] = -12 * (-2\/3)x^(-5\/3) = 8x^(-5\/3)<\/p>\n\n\n\n Now evaluate at x = 8: Evaluate at x = -8: f(8) = 3(8) – 36(8)^(1\/3) = 24 – 36(2) = 24 – 72 = -48 Relative maximum: (x, y) = (-8, 48) To find the relative extrema of the function f(x) = 3x – 36x^(1\/3), we begin by taking the derivative to identify critical points. The derivative f\u2032(x) = 3 – 12x^(-2\/3) represents the slope of the tangent line. We set the derivative equal to zero to locate any stationary points. Solving the equation 3 – 12x^(-2\/3) = 0 yields x = \u00b18.<\/p>\n\n\n\n Once the critical points are found, we determine whether they are maxima or minima. This is done using the second derivative test. Taking the second derivative, we get f\u2033(x) = 8x^(-5\/3). We substitute the critical points into this second derivative. For x = 8, f\u2033(8) = 1\/4, which is positive. This indicates that the function is concave up at x = 8, confirming a relative minimum. For x = -8, f\u2033(-8) is negative, so the function is concave down, confirming a relative maximum.<\/p>\n\n\n\n To complete the analysis, we plug the x-values back into the original function to find the corresponding y-values. f(8) = -48 and f(-8) = 48. These are the coordinates of the relative minimum and maximum, respectively.<\/p>\n\n\n\n Understanding relative extrema helps describe where the function increases or decreases most significantly. The relative maximum at (-8, 48) shows a local peak, while the relative minimum at (8, -48) shows a local dip in the graph. These points are important in applications like optimization and graph sketching.<\/p>\n\n\n\n Find all relative extrema of the function. (If an answer does not exist, enter DNE:) f(x) = 3x – 36x^(1\/3) relative maximum (x, Y) = relative minimum (x, Y) The Correct Answer and Explanation is: To find the relative extrema of the functionf(x) = 3x – 36x^(1\/3) Step 1: Find the first derivative We use […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-34996","post","type-post","status-publish","format-standard","hentry","category-quiz-questions"],"_links":{"self":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/34996","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/comments?post=34996"}],"version-history":[{"count":1,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/34996\/revisions"}],"predecessor-version":[{"id":35002,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/34996\/revisions\/35002"}],"wp:attachment":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/media?parent=34996"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/categories?post=34996"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/tags?post=34996"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
f(x) = 3x – 36x^(1\/3)<\/strong><\/p>\n\n\n\nStep 1: Find the first derivative<\/h3>\n\n\n\n
f\u2032(x) = d\/dx[3x] – d\/dx[36x^(1\/3)]
f\u2032(x) = 3 – 36 * (1\/3)x^(-2\/3)
f\u2032(x) = 3 – 12x^(-2\/3)<\/p>\n\n\n\nStep 2: Set the derivative equal to zero<\/h3>\n\n\n\n
3 = 12x^(-2\/3)
Divide both sides by 12:
1\/4 = x^(-2\/3)
Now solve for x:<\/p>\n\n\n\n
Take the reciprocal:
x^(2\/3) = 4
Raise both sides to the 3\/2 power:<\/p>\n\n\n\nStep 3: Use the second derivative test or first derivative sign chart<\/h3>\n\n\n\n
Differentiate again:<\/p>\n\n\n\n
f\u2033(8) = 8*(8)^(-5\/3) = 8 \/ (8^(5\/3))
= 8 \/ [2^5] = 8 \/ 32 = 1\/4 > 0
So f has a relative minimum<\/strong> at x = 8<\/p>\n\n\n\n
f\u2033(-8) = 8 * (-8)^(-5\/3)
Negative base to an odd power gives negative result:
f\u2033(-8) < 0
So f has a relative maximum<\/strong> at x = -8<\/p>\n\n\n\nStep 4: Find the y-values<\/h3>\n\n\n\n
f(-8) = 3(-8) – 36(-8)^(1\/3) = -24 – 36(-2) = -24 + 72 = 48<\/p>\n\n\n\nFinal Answer:<\/h3>\n\n\n\n
Relative minimum: (x, y) = (8, -48)<\/p>\n\n\n\n
\n\n\n\nExplanation <\/h3>\n\n\n\n
![\"\"](\"https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-614.jpeg\")
<\/figure>\n","protected":false},"excerpt":{"rendered":"