Math  /  Calculus

QuestionConsider the function defined by f(x)=x4+15x36f(x)=\sqrt{x^{4}+15}-\frac{x^{3}}{6}
FORMATTING for this question: If there is more than one answer, separate them with a semi-colon (;). If there are none, write none. Give exact values. (a) Provide a list of all the critical numbers of ff. x=x= \square (b) Provide a list of all the local minima of ff x=x= \square (1) (c) Provide a list of all the local maxima of ff. x=x=\square

Studdy Solution
Use the first derivative test to classify the critical numbers as local minima or maxima.
Evaluate the sign of f(x) f'(x) around each critical number:
- For x=15 x = -\sqrt{15} : Check intervals (,15) (-\infty, -\sqrt{15}) and (15,1) (-\sqrt{15}, -1) . - For x=1 x = -1 : Check intervals (15,1) (-\sqrt{15}, -1) and (1,1) (-1, 1) . - For x=1 x = 1 : Check intervals (1,1) (-1, 1) and (1,15) (1, \sqrt{15}) . - For x=15 x = \sqrt{15} : Check intervals (1,15) (1, \sqrt{15}) and (15,) (\sqrt{15}, \infty) .
Determine whether f(x) f'(x) changes from positive to negative (local max) or negative to positive (local min).
- x=1 x = -1 is a local maximum. - x=1 x = 1 is a local minimum.
(a) Critical numbers of f f : x=15;1;1;15 x = -\sqrt{15}; -1; 1; \sqrt{15}
(b) Local minima of f f : x=1 x = 1
(c) Local maxima of f f : x=1 x = -1

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