Question

Investigate the family of curves given by

\(f\left( x \right) = {x^4} + {x^3} + c{x^2}\)

In particular you should determine the transitional value ofc at which the number of critical number changes and the transitional value at which the number of inflection points changes. Illustrate the various possible shapes with graphs.

Short answer

If \(c < 9/32\) , then there are total 3 critical numbers.

If \(c = 9/32\) , then there are total 2 critical numbers.

If \(c > 9/32\) , then there are total 1 critical numbers.

If \(c \ge 6/16\) , then there are no inflection points.

If \(c < 6/16\) , then there are total 2 inflection numbers.

The required graph is,

Step-by-step solution

Critical number

Given that \(f\left( x \right) = {x^4} + {x^3} + c{x^2}\).

\(\begin{aligned}{c}f'\left( x \right) = 4{x^3} + 3{x^2} + 2cx\\ = x\left( {4{x^2} + 3x + 2c} \right)\end{aligned}\).

Critical number occurs when \(f'\left( x \right) = 0\).

That is \(x = 0\)

Or \(4{x^2} + 3x + 2c = 0\)

The solution of the quadratic equation depends upon the discriminant

\(\begin{aligned}{c}D = \sqrt {9 - 4\left( 4 \right)\left( {2c} \right)} \\ = \sqrt {9 - 32c} \end{aligned}\)

So,

If \(c < 9/32\) , then the quadratic equation has two roots.

Hence, there are total 3 critical numbers.

If \(c = 9/32\) , then The quadratic equation has 1 root.

Hence, there are total 2 critical numbers.

If \(c > 9/32\) , then the quadratic equation has no roots.

Hence, there are total 1 critical numbers.

Inflection points

Now, \(f'\left( x \right) = 4{x^3} + 3{x^2} + 2cx\)

So \(f''\left( x \right) = 12{x^2} + 6x + 2c\)

And \(f'''\left( x \right) = 24x + 6\)

Put \(f'''\left( x \right) = 0\)

So \(x = - \frac{1}{4}\)

\(f''\left( { - \frac{1}{4}} \right) = \frac{{16c - 16}}{{16}}\)

If \(c \ge 6/16\) , then \(f''\left( { - \frac{1}{4}} \right) \ge 0\). Hence, there are no inflection points.

If \(c < 6/16\) , then \(f''\left( { - \frac{1}{4}} \right) < 0\). Hence, there are total 2 inflection numbers.

Illustrate family of graphs

The family of graph is as follows:

Conclusion

If \(c < 9/32\) , then there are total 3 critical numbers.

If \(c = 9/32\) , then there are total 2 critical numbers.

If \(c > 9/32\) , then there are total 1 critical numbers.

If \(c \ge 6/16\) , then there are no inflection points.

If \(c < 6/16\) , then there are total 2 inflection numbers.