hellp me to explain..!!11

Question

Explain IN DETAIL why a cubic function with 1 turning point does not exist.

Answer 1

A polynomial function f(x) = a_nxⁿ + a_n-1x^n-1 + a_n-2x^n-2 +....+ a₂x² + a₁x + a₀

A turning point of a function is a point where the graph of a function changes from sloping downwards to sloping upwards, or vice versa.So the gradient changes from negative to positive or from positive to negative.

Depending up on the Degrees of polynomial functions we determine the possible turning points.

The graph of polynomial of degree n has at most n - 1 turning points.

The graph of a polynomial of even degree has an odd number of turning points,while the graph of a polynomial of odd degree has an even number of turning points.

 

For example, a quadratic function y = ax² + bx + c

Degree(n) = 2

Turning points (n - 1) = 2 - 1 = 1

Number turning points of quadratic function = 1.

 

And now cubic function f(x) = ax³ + bx² + cx + d

Degree(n) = 3

Turning points (n - 1) = 3 - 1 = 2

However, some cubics have fewer turning points for example f(x) = x³,But no cubic have more than two turning points.

Number of turning points of cubic function 0 or 2.

Therefore, a cubic function 1 turning point does not exist.