Question

Show that the function f(x) = x 3 + x 2 + 5x + 6 is an increasing function.

Answer 1

Answer:

So, it is proved that f(x) is an increasing function.

Step-by-step explanation:

f(x) = x^3+x^2+5x+6

Now put x = 1,

f(1) = (1)^3+(1)^2+5(1)+6

=1+1+5+6

=13

Now put x = 2,

f(2) = (2)^3+(2)^2+5(2)+6

=8+4+10+6

=28

Now put x = 0,

f(0) = (0)^2+(0)^2+5(0)+6

=0+0+0+6

=6

So, here we observe as the values of x increase the value of f(x) increases and as we decrease it decreases.

So, it is proved that f(x) is an increasing function.

Answer 2

$\large\underline{\sf{Solution-}}$

Given that,

$\rm :\longmapsto\:f(x) = {x}^{3} + {x}^{2} + 5x + 6$

On differentiating both sides w. r. t. x, we get

$\rm :\longmapsto\: \: \dfrac{d}{dx} f(x) = \dfrac{d}{dx}( {x}^{3} + {x}^{2} + 5x + 6)$

$\rm :\longmapsto\: \: f'(x) = \dfrac{d}{dx}{x}^{3} + \dfrac{d}{dx}{x}^{2} +\dfrac{d}{dx} 5x + \dfrac{d}{dx}6$

We know,

$\green{ \boxed{ \bf \: \dfrac{d}{dx} {x}^{n} = {nx}^{n - 1}}}$

So, using this

$\rm :\longmapsto\:f'(x) = {3x}^{2} + 2x + 5 + 0$

$\rm :\longmapsto\:f'(x) = {3x}^{2} + 2x + 5$

Now,

We know,

$\rm :\longmapsto\:In \: a \: quadratic \: expression \: {ax}^{2} + bx + c, \: if \:$

$\rm :\longmapsto\:a > 0 \: and \: discriminant \: < 0 \: then \: {ax}^{2} + bx + c > 0$

Here,

Quadratic expression is .

$\rm :\longmapsto\: {3x}^{2} + 2x + 5$

Here,

$\rm :\longmapsto\:a = 3$

$\rm :\longmapsto\:b = 2$

$\rm :\longmapsto\:c = 5$

So,

$\rm :\longmapsto\:Discriminant = {b}^{2} - 4ac$

$\rm  \:  =  \: \: {(2)}^{2} - 4 \times 3 \times 5$

$\rm  \:  =  \: \:4 - 60$

$\rm  \:  =  \: \: - 54$

So, we have now,

$\rm :\longmapsto\:a = 3 > 0 \: \: and \: Discriminant = - 54 < 0$

$\rm :\implies\: {3x}^{2} + 2x + 5 > 0$

$\bf\implies \:f'(x) > 0$

$\bf\implies \:f(x) \: is \: always \: increasing \: for \: every \:x \in \: R$

Additional Information :-

↝ If the functions f and g are increasing (or decreasing) on the interval (a,b), then the sum of the functions f+g is also increasing (or decreasing) on this interval.

↝ If the function f(x) is increasing (or decreasing) on the interval (a,b), then the opposite function − f(x)  is decreasing (or increasing) on this interval.

↝ If the function f(x) is increasing (or decreasing) on the interval (a,b), then the inverse function of f(x) is decreasing (or increasing) on this interval.