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Let f be a non constant polynomial of degree k and let g:R→RR\to RR→R be a bounded continuous functionWhich of the following statements are neces

Question

Let f be a non constant polynomial of degree k and let g: $R\to R$ be a bounded continuous function
Which of the following statements are necessarily true?

There always exists $x0∈Rx_0 \in \mathbb{R}$ such that $f(x_0) = g(x_0)$ .

There exist no $x_0 \in \mathbb{R}$ such that $f(x_0)=g(x_0)$ .

There exists $x_0 \in \mathbb{R}$ such that $f(x_0)=g(x_0)$ if k is even.

Solution

Correct option is D

$\textbf{Intermediate value theorem:}$

The Intermediate Value Theorem (IVT) states that if a continuous function takes two values at some points

and the function is continuous between these points, then it must take any value between them at some point in that interval.

Let,

Now Let $f(x)=x^k\\[10pt]and,h(x)=f(x)-g(x)\\[10pt]\text{if k=odd:}\\[10pt] \lim_{n\to-\infty}[x^k-g(x)] =-\infty\ (\because \text{g is bounded}) \cdots(1)\\[10pt]\lim_{n\to \infty}[x^k-g(x)]=\infty , \text{and h(x) is continuous in R } \cdots(2)\\[10pt]$

Applying I.V.P we can say that h(x) acuires all values in R.

$\implies \exist x_0\in R:h(x)=0 \ \text{if k is odd}\\[10pt]\implies \exist x_0 \in R:f(x_0)=g(x_0)\ \text{if k is odd.}\\[10pt]\textbf{Option D is correct.}$

And if k is even then both the limits in (1) and (2) will be $\infty$ .

and we will not be able to apply I.V.P so all other statements are not necessarily true.

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Let f be a non constant polynomial of degree k and let g: $R\to R$ be a bounded continuous function
Which of the following statements are necessarily true?

There always exists $x0∈Rx_0 \in \mathbb{R}$ such that $f(x_0) = g(x_0)$ .

There exist no $x_0 \in \mathbb{R}$ such that $f(x_0)=g(x_0)$ .

There exists $x_0 \in \mathbb{R}$ such that $f(x_0)=g(x_0)$ if k is even.

There exists $x_0 \in \mathbb{R}$ such that $f(x_0)=g(x_0)$ if k is even.

Correct option is D

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Let f be a non constant polynomial of degree k and let g:R→RR\to RR→R be a bounded continuous functionWhich of the following statements are necessarily true?

There always exists x0∈Rx_0 \in \mathbb{R}x0_00​​​∈R such that f(x0)=g(x0)f(x_0) = g(x_0)f(x0_00​​​)=g(x0_00​​​).​

There exist no x0∈Rx_0 \in \mathbb{R}x0​∈R​ such thatf(x0)=g(x0) f(x_0)=g(x_0)f(x0​)=g(x0​).​

There existsx0∈R x_0 \in \mathbb{R}x0​∈R​ such thatf(x0)=g(x0) f(x_0)=g(x_0)f(x0​)=g(x0​)​ if k is even.

There exists x0∈Rx_0 \in \mathbb{R}x0​∈R​ such that f(x0)=g(x0)f(x_0)=g(x_0)f(x0​)=g(x0​)​ if k is even.

Correct option is D

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Let f be a non constant polynomial of degree k and let g: $R\to R$ be a bounded continuous function
Which of the following statements are necessarily true?

There always exists $x0∈Rx_0 \in \mathbb{R}$ such that $f(x_0) = g(x_0)$ .

There exist no $x_0 \in \mathbb{R}$ such that $f(x_0)=g(x_0)$ .

There exists $x_0 \in \mathbb{R}$ such that $f(x_0)=g(x_0)$ if k is even.

There exists $x_0 \in \mathbb{R}$ such that $f(x_0)=g(x_0)$ if k is even.