GATE CSE 1987 | Question: 1-xxvi

Question

If $f(x_{i}).f(x_{i+1})< 0$ then

• A. There must be a root of $f(x)$ between $x_i$ and $x_{i+1}$
• B. There need not be a root of $f(x)$ between $x_{i}$ and $x_{i+1}$
• C. There fourth derivative of $f(x)$ with respect to $x$ vanishes at $x_{i}$
• D. The fourth derivative of $f(x)$ with respect to $x$ vanishes at $x_{i+1}$

Comments

The answer should be B as the graph could be discontinuous and B is the most suitable answer. Options C and D are not understood. Anybody, please explain those options!

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what is the significance of 4^th derivative anyway?

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What if the function is (-1)^x. Then the multiplication will be < 0 and still it won’t cut the X-axis.

Answer 1

As $f(x_{i}).f(x_{i+1}) <0$

Means one of them is positive and one of them in negative . as their multiplication is negative.

So, when you draw the graph for $f(x)$ where $x_{i}\leq x \leq x_{i+1} $. Definitely $F(x)$ will cut the $X$- axis.

So, there will definitely a root of $F(x)$ between $x_{i}$ and $x_{i+1}.$

Correct Answer: A.

Comments

@Verma Ashish

Yeah, if we assume $f$ as discontinuous function then (B) will be correct. As it is very old gate question, So, I think, some information is missing in question.

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@Abhrajyoti00 @Udhay_Brahmi @jiren waht is significance of 4^th derivative here?

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We talk about significance of something with respect to a statement when that statement is true but here statements C and D are not correct because consider $f(x) = (x- 1)^5$ and $x_1=\frac{1}{2}$ and $x_2=\frac{3}{2}.$ So, here $f(\frac{1}{2}) f(\frac{3}{2}) < 0$  But neither $f^{(4)} (\frac{1}{2}) = 0$ nor $f^{(4)} (\frac{3}{2}) = 0.$

Answer 2

here ,option A is true because ,

f(x_i) * f(x_i+1) < 0 ; which means one of them should be negative i.e both must have opposite sign,  

and if there exist the opposite that means they will cut the x-axis and when they cut the x axis then there exist roots we all know that