Explanation Consider the function f ( x ) = x 4 + x 3 + c x 2 . Obtain the first derivative of f ( x ) f ′ ( x ) = d d x ( f ( x ) ) = d d x ( x 4 + x 3 + c x 2 ) = 4 x 3 + 3 x 2 + 2 c x = x ( 4 x 2 + 3 x + 2 c ) Obtain the critical point to the function. Equate the f ′ ( x ) = 0 x ( 4 x 2 + 3 x + 2 c ) = 0 4 x 2 + 3 x + 2 c = 0 This is quadratic equation to find the roots of the equation by using roots = − b ± b 2 − 4 a c 2 a x = − 3 ± ( 3 ) 2 − 4 ⋅ 4 ⋅ 2 c 2 ⋅ 4 = − 3 ± 9 − 32 c 8 Thus, x = − 3 ± 9 − 32 c 8 (1) If determinant b 2 − 4 a c = 9 − 32 c is 0 then the value of c is calculated as follows, 9 − 32 c = 0 − 32 c = − 9 c = 9 32 Substitute c = 9 32 in equation (1), x = − 3 ± 9 − 32 ( 9 32 ) 8 = − 3 8 Therefore, there are two critical points in the equation x = − 3 8 and 0 if c = 9 32 . If determinant is negative then, negative occurs in root. The only critical point at x = 0 . If 0 < c < 9 32 there are three critical points then x = − 3 + 9 − 32 c 8 , − 3 − 9 − 32 c 8 and 0 Obtain the second derivative of f ( x ) f " ( x ) = d d x ( f ′ ( x ) ) = d d x ( 4 x 3 + 3 x 2 + 2 c x ) = 12 x 2 + 6 x + 2 c Thus, f " ( x ) = 12 x 2 + 6 x + 2 c (2) Substitute x = 0 in equation (2), f " ( 0 ) = 12 ( 0 ) 2 + 6 ( 0 ) + 2 c = 2 c Since 0 < c < 9 32 , The second derivative f " ( 0 ) is positive. Thus, the function f ( x ) is local minimum at x is zero. Substitute x = − 3 − 9 − 32 c 8 in equation (2), f " ( − 3 − 9 − 32 c 8 ) = 12 ( − 3 − 9 − 32 c 8 ) 2 + 6 ( − 3 − 9 − 32 c 8 ) + 2 c Since 0 < c < 9 32 , The second derivative f " ( 0 ) is positive

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Chapter 4, Problem 76E

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To investigate: The family of curves of the function f(x)=x4+x3+cx2 .

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Chapter 4, Problem 75E

Chapter 4, Problem 77E

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Ch. 4.1 - Explain the difference between an absolute minimum...Ch. 4.1 - Suppose f is a continuous function defined on a...Ch. 4.1 - For each of the numbers a, b, c, d, r, and s,...Ch. 4.1 - For each of the numbers a, b, c, d, r, and s,...Ch. 4.1 - Use the graph to state the absolute and local...Ch. 4.1 - Use the graph to state the absolute and local...Ch. 4.1 - Sketch the graph of a function f that is...Ch. 4.1 - Sketch the graph of a function f that is...Ch. 4.1 - Sketch the graph of a function f that is...Ch. 4.1 - Sketch the graph of a function f that is...

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To investigate: The family of curves of the function f ( x ) = x 4 + x 3 + c x 2 .

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To investigate: The family of curves of the function f(x)=x4+x3+cx2 .

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