1. Tardigrade
  2. Question
  3. Mathematics
  4. Consider the function f(x)= max x2,(1-x)2, 2 x(1-x) where 0 ≤ x ≤ 1. Let Rolle's Theorem is applicable for f(x) on greatest interval [a, b] then a+b+c is (where c is point such that f prime(c)=0 )

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Q. Consider the function $f(x)=\max \left\{x^2,(1-x)^2, 2 x(1-x)\right\}$ where $0 \leq x \leq 1$. Let Rolle's Theorem is applicable for $f(x)$ on greatest interval $[a, b]$ then $a+b+c$ is (where $c$ is point such that $f^{\prime}(c)=0$ )

Application of Derivatives

Solution:

Draw the graph of $f_1(x)=x^2, f_2(x)=(1-x)^2 \& f_3=2 x(1-x)$
Now the bold part is the graph of $f(x)$
Hence
image
$f(x)=\begin{cases} (1-x)^2 & , 0 \leq x < \frac{1}{3} \\ 2 x(1-x) & , \frac{1}{3} \leq x \leq \frac{2}{3} \\ x^2 & , \frac{2}{3} < x \leq 1 \end{cases}$
Clearly Rolle's theorem is applicable on $\left[\frac{1}{3}, \frac{2}{3}\right]$
where $ f(x)=2 x(1-x) \Rightarrow f^{\prime}(c)=2-4 c=0 \Rightarrow c=1 / 2 $
$ \Rightarrow a+b+c=\frac{1}{3}+\frac{2}{3}+\frac{1}{2} \Rightarrow \frac{3}{2} $