Find a, b, c, and d such that the cubic function f(x) = ax3 + bx? + cx + d satisfies the given conditions. Relative maximum: (2,9)
Relative minimum: (4,3)
Inflection point: (3,6)
a =
b =
C=
d =

Where:

Step-by-step explanation:

We are given a cubic function:

And we want to find a, b, c and d such that the  function has a relative maximum at (2, 9); a relative mininum at (4, 3); and an inflection point at (3, 6).

Since the function has a relative maximum at (2, 9), this means that:

Simplify:

Likewise, since it has a relative minimum at (4, 3):

Simplify:

We can subtract the first equation from the second. So:

Simplify:

Divide both sides by two. Hence:

Relative minima occurs only at the critical points of a function. That is, it occurs whenever the first derivative equals zero.

Find the first derivative. We can treat a, b, c and d as constant. Hence:

Since it has a minima at (2, 9), it means that:

Thus:

(We will only need one of the two points to complete the problem.)

Inflection points occurs whenever the second derivative of a function equals zero. Find the second derivative:

Since there is a inflection point at (3, 6):

Solve for b:

Substitute this into the above equation:

Solve for c:

Substitute b and c into the previously acquired equation:

Solve for a:

Solve for b and c:

Using either the very first or second equation, solve for d:

Hence:

Hence, our function is:

22 dimes and 9 nickels

step-by-step explanation: