Question

Quartic Polynomial Functions Let \(f(x)=\) \(a x^{4}+b x^{3}+c x^{2}+d x+e\) with \(a \neq 0\) (a) Show that the graph of \(f\) has 0 or 2 points of inflection. (b) Write a condition that must be satisfied by the coefficients if the graph of \(f\) has 0 or 2 points of inflection.

Short answer

The graph of a quartic function has either 0 or 2 points of inflection. The conditions to determine the number of inflection points are derived from the discriminant of the second derivative. If the discriminant of the quadratic function \(12ax^{2}+6bx+2c=0\) is negative (i.e., \(b^2 - 4ac < 0\)), then there are 0 inflection points. If the discriminant is zero or positive (i.e., \(b^2 - 4ac \geq 0\)), then there are 2 inflection points.

Step-by-step solution

Compute First derivative of f(x)

The first derivative of the function \(f(x)\) is given by \(f'(x) = 4ax^{3}+3bx^{2}+2cx+d\). This derivative tells us the rate of change of the function.

Compute Second derivative of f(x)

The second derivative of the function \(f(x)\) is given by \(f''(x) = 12ax^{2}+6bx+2c\). This second derivative tells us if the function is concave up or down.

Solving f''(x) for zeros

In order to determine the points of inflection, we set the second derivative equation equals to zero, \(12ax^{2}+6bx+2c=0\). This gives us the 'x' values at which the inflection points may appear.

Conditions determining number of Inflection Points

The discriminant of the quadratic equation \(12ax^{2}+6bx+2c=0\) is given by \(D = b^2 - 4ac\). If \(D < 0\), the equation has no roots which mean \(0\) inflection points. If \(D \geq 0\), the equation has one or two roots which means \(2\) inflection points.