If f(x) = x^3 - 10x^2 + 200x - 10,then | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) Given $f(x) = x^3 - 10x^2 + 200x - 10$.To determine the intervals of increase or decrease,we find the derivative $f'(x)$:$f'(x) = 3x^2 - 20x + 200

Vedclass · Accepted Answer

$f(x)$ is increasing throughout the real line

Vedclass · Answer

(C) Given $f(x) = x^3 - 10x^2 + 200x - 10$.To determine the intervals of increase or decrease,we find the derivative $f'(x)$:$f'(x) = 3x^2 - 20x + 200$.For the function to be increasing,we require $f'(x) > 0$. Consider the quadratic expression $3x^2 - 20x + 200$.The discriminant $D$ of this quadratic is $D = b^2 - 4ac = (-20)^2 - 4(3)(200) = 400 - 2400 = -2000$.Since $D  0)$,the quadratic $3x^2 - 20x + 200$ is always positive for all real values of $x$.Alternatively,by completing the square:$f'(x) = 3(x^2 - \frac{20}{3}x) + 200 = 3(x^2 - \frac{20}{3}x + \frac{100}{9} - \frac{100}{9}) + 200 = 3(x - \frac{10}{3})^2 - \frac{100}{3} + 200 = 3(x - \frac{10}{3})^2 + \frac{500}{3}$.Since $3(x - \frac{10}{3})^2 \ge 0$ for all $x \in \mathbb{R}$,it follows that $f'(x) \ge \frac{500}{3} > 0$ for all $x \in \mathbb{R}$.Therefore,$f(x)$ is strictly increasing throughout the real line.

If $f(x) = x^3 - 10x^2 + 200x - 10$,then

Similar Questions

Give various causes for seed dormancy.

The reading of a spring balance corresponds to $100 N$ while situated at the north pole and a body is kept on it. The weight record on the same scale if it is shifted to the equator, is $...........\,N$

(take, $g =10 m / s ^2$ and radius of the earth, $R =6.4 \times 10^3 m$ )

The common bottle cork is a product of .......... .

If $f_n(x) = \log \log \log \ldots \log x$ (where $\log$ is repeated $n$ times),then $\int (x f_1(x) f_2(x) \ldots f_n(x))^{-1} dx$ is equal to

The correct order of reactivity of $PhMgBr$ with: $(I)$ $Ph-CO-Ph$,$(II)$ $CH_3-CHO$,$(III)$ $CH_3-CO-CH_3$ is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module