Question

f(x)=x⁷+5x³+125 is......,Select correct option from the given options.
(a) decreasing in (0,∞)
(b) decreasing in (-∞, 0)
(c) increasing on R
(d) neither increasing nor decreasing in R

Answer 1

function, f(x) is increasing in interval (a,b) only when f'(x) > 0 within (a, b)
and decreasing only when f'(x) < 0 within (a,b)

here, f(x) = x⁷+5x³+125

differentiate , $f'(x)=7x^6+15x^2$

$f'(x)=x^2(7x^4+15)$

we see, for any real value of x, f'(x) > 0

e.g., f'(x) >0 $\forall x\in\mathbb{R}$
therefore f(x) is increasing for all real numbers .

hence, option (c) is correct choice .

Answer 2

Hello,

Solution:

Steps to find out increasing or decreasing function:

1) differentiate the given function with respect to x,

2) check in given interval ,the value of function becomes positive or negative

3) if value becomes positive then function is increasing otherwise

$f(x) = {x}^{7} + 5 {x}^{3} + 125 \\ \\ \frac{dy}{dx} = 7 {x}^{6} + 15 {x}^{2} + 0 \\ \\ \frac{dy}{dx} = {x}^{2} (7 \: {x}^{4} + 15)$
Now, as we can see that obtained expression always achieve positive value in the entire range of R,because of even powers of x.

So, the given function is increasing on R (Option C)

Hope it helps you.