Theory of Equations 5 Question 13

13. The function $f^{\prime}(x)$ is

(a) increasing in $-t,-\frac{1}{4}$ and decreasing in $-\frac{1}{4}, t$

(b) decreasing in $-t,-\frac{1}{4}$ and increasing in $-\frac{1}{4}, t$

(c) increasing in $(-t, t)$

(d) decreasing in $(-t, t)$

Passage II

If a continuous function $f$ defined on the real line $R$, assumes positive and negative values in $R$, then the equation $f(x)=0$ has a root in $R$. For example, if it is known that a continuous function $f$ on $R$ is positive at some point and its minimum values is negative, then the equation $f(x)=0$ has a root in $R$. Consider $f(x)=k e^{x}-x$ for all real $x$ where $k$ is real constant.

(2007, 4M)

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Answer:

Correct Answer: 13. (b)

Solution:

  1. As, $f^{\prime \prime}(x)=2(12 x+3)$

$f^{\prime}(x)>0$, when $x>-\frac{1}{4}$ and

$f^{\prime}(x)<0$, when $x<-\frac{1}{4}$.

$\therefore$ It could be shown as



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