1 Marks Question

Question 11 Mark

Prove that the logarithmic function is increasing on $(0, \infty)$.

Answer

Let$
f(x)=\log _e x, x>0
$
$\therefore \quad \quad f^{\prime}(x)=\frac{1}{x}=+ ve$, for $x>0$
$\Rightarrow$ for $x \leftarrow(0, \infty), f^{\prime}(x)>0$
Therefore, the given logarithmic function is increasing in $(0, \infty)$

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Question 21 Mark

What different values of $a$ function $f(x)=a x+b$ is decreasing when $x \in R$.

Answer

$
\begin{aligned}
f(x) & =a x+b \\
\Rightarrow \quad f^{\prime}(x) & =a
\end{aligned}
$
$f$ is decreasing if $f^{\prime}(x)<0$
or$\quad$$
a<0
$
hence for $a<0$ function will decreasing

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Question 31 Mark

Show that function $f(x)=7 x^2-3, x>0$ is increasing function.

Answer

$
f(x)=7 x^3-3, x>0
$
then$
f^{\prime}(x)=14 x-0
$$\forall x>0$
$\Rightarrow \quad f^{\prime}(x)>0, \quad \forall x>0$
hence function $f(x)$ is increasing function because $f^{\prime}(x)>0$.

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Question 41 Mark

What do you mean by differential coefficient $\frac{d y}{d x}$ ?

Answer

Differential coefficient $\frac{d y}{d x}$ is tangent of that angle which makes tangent line $x$-axis at any point $P (x$, $y$ ) of curve $y=f(x)$. Generally it is known as gradient of curve at point P .

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