A curve with equation \(y=ax^{3} +bx\) passes through the origin at an angle of \(45° \)

Passes through the point \(( 2 , - 6 ) \)

a) Find the values of \(a\) and \(b\).

Question

A curve with equation \(y=ax^{3} +bx\) passes through the origin at an angle of \(45° \)

Passes through the point \(( 2 , - 6 ) \)

a) Find the values of \(a\) and \(b\).

Answer

\(a=-1, b=1\)

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Follow the steps below to find the nonnegative numbers \( x \) and \( y \) that satisfy the given requirements. Give the optimum value of the indicated expression. Complete parts (a) through (f) below.

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(d) Find \( \frac { d P } { d x } \) . Solve the equation \( \frac { d P } { d x } = 0 \) .

\( \frac { d P } { d x } = 170 - 2 x \) and when \( \frac { d P } { d x } = 0 , x = 85 \) (Use a comma to separate answers as needed.)

(e) Evaluate \( P \) at any solutions found in part (d), as well as at the endpoints of the domain found in part (c).

To answer the first part, select the correct choice below and, if necessary, fill in the answer box(es) within your choice.

A. There was one solution in part (d). For that solution, \( P = 7225 \) .

B. There were two solutions in part (d). For the lesser value of \( x , P = \) . For the greater value of \( x , P = \)

Evaluate \( P \) at the endpoints of the domain. At the lower endpoint, \( P = \square \) . At the upper endpoint, \( P = \square \) .

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Consider the function \( f ( x ) = 5 x - 8 \) and find the following: a) The average rate of change between the points \( ( - 1 , f ( - 1 ) ) \) and \( ( 4 , f ( 4 ) ) \) .

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Using the given graph of the function \( f \) , find the following.

(a) the intercepts, if any

(b) its domain and range

(c) the intervals on which it is increasing, decreasing, or constant

(d) whether it is even, odd, or neither