TIFR CSE 2015 | Part A | Question: 11

Question

Suppose that $f(x)$ is a continuous function such that $0.4 \leq f(x) \leq 0.6$ for $0 \leq x \leq 1$. Which of the following is always true?

• A. $f(0.5) = 0.5$.
• B. There exists $x$ between $0$ and $1$ such that $f(x) = 0.8x$.
• C. There exists $x$ between $0$ and $0.5$ such that $f(x) = x$.
• D. $f(0.5) > 0.5$.
• E. None of the above statements are always true.

Comments

Why answer is B here ? Can anyone prove it ?

Answer 1

(A) f(0.5)=0.5, we cannot say here f(x) value always trueBecause we need to know f(x) value between

0.4≤ f(x) ≤ 0.6, and here we are getting f(x) value when x=0.5

(C)Here we know f(x) value between 0 to 0.5. But when f(x)=0.6 , x value may be ≥1

(D) Here also we cannot predict f(x) value when 0.4≤ f(x) ≤ 0.6 

f(0.5)>0.5 is an inequality. So, we cannot get any exact value of x

Now for (B) Here we can see the f(x) value 0.4≤ f(x) ≤ 0.6 when x  between 0 to 1

for eg: f(0.5)=0.4, where x value is 0.5

            f(0.6)=0.48,where x value is 0.6

            f(0.7)=0.56 , where x value is 0.7

here we are only concern about f(x) is between 0.4 and 0.6.

so, here value of x always between 0 ≤ x ≤ 1 when 0.4≤ f(x) ≤ 0.6

So, answer will be (B)

Comments

you haven't proved the existence of such an $x$.

---

then whats the answer sir ? @ Arjun

---

option B is correct only, I have given answer now.

---

@srestha  ma,am you had not provided any reason to eliminate e, you are choosing b by eliminating a,c,d.