Question

Prove that the function \(f(x) = {x^{101}} + {x^{51}} + x + 1\) has neither a local maximum nor a local minimum.

Short answer

The function \(f(x) = {x^{101}} + {x^{51}} + x + 1\) has neither a local maximum nor a local minimum.

Step-by-step solution

Given function

The function is \(f(x) = {x^{101}} + {x^{51}} + x + 1\).

Definition of Local minimum and Local Maximum

The local minimum is a point within an interval at which the function has a minimum value.

The relative minima is the minimum point in the domain of the function, with reference to the points in the immediate neighbourhood of the given point.

The local maximum is a point within an interval at which the function has a maximum value.

The absolute maxima is also called the global maxima and are the point across the entire domain of the given function, which gives the maximum value of the function.

Check the function for the Local Minimum and Maximum

The function is\(f(x) = {x^{101}} + {x^{51}} + x + 1\).

Obtain the first derivative of\(f(x)\).

\({f^\prime }(x) = 101{x^{100}} + 51{x^{50}} + 1\)

Observe that\({f^\prime }(x)\)will be positive for any value of\(x\)as the polynomial has no negative sign and the even powers.

So, the graph of\(f(x)\)must be an increasing graph.

Thus, it can be concluded that the function has no local extreme as the increasing graph does not have local extreme.

Therefore, the function \(f(x) = {x^{101}} + {x^{51}} + x + 1\) has neither a local maximum nor a local minimum.