Question

The statement, 'A polynomial function is continuous for all real numbers" isA true for all polynomial functions.B. true for some polynomial functions.Oc.C. never true for polynomial functions.

Answer 1

The statement is given ''A polynomial function is continuous for all real numbers" .

Consider the polynomial

`f(x)=a_0+a_1x+_{}a_1x^2\ldots\ldots\ldots\ldots\ldots\ldots\ldots\text{.}\mathrm{}a_nx^n_{}`

Since every polynomial function is valid for every rela number.

Prove continuity for the polynomial function at any point c.

`\lim _(x\rightarrow c)f(x)=f(c)`

For LHS,

`\lim _(x\rightarrow c)f(x)=\lim _(x\rightarrow c)(a_0+a_1x+\ldots\ldots\ldots\ldots a_nx^n)`

Susbtitute x=c.

`a_0+a_1c_{}+\ldots\ldots\ldots\ldots\ldots\ldots\ldots.a_nc^n`

For RHS

`f(c)=a_0+a_1c+\ldots\ldots\ldots\ldots\ldots\ldots..a_nc^n`

Then LHS=RHS.

The function is continuous at x=c.

Hence every polynomial function is continuous for all real numbers.

The correct option is A.