If {f}(x)=x³-10 x²+200 x-10, then

Correct Answer is : f(x){f}(x)f(x) is increasing throughout real line

{f}(x) is decreasing in (-∞, 10] and increasing in [10, ∞)

f(x) is increasing in (-∞, 10] and decreasing in [10, ∞)

{f}(x) is increasing throughout real line

{f}(x) is decreasing throughout real line

Correct Answer is : f(x){f}(x)f(x) is increasing throughout real line

Test Index

MHTCET 15 May 2024 Shift 1 Paper

Question : 118 of 150

Marks: +1, -0

\mathrm{f}(x)=x^{3}-10 x^{2}+200 x-10

, then

$\mathrm{f}(x)$ is decreasing in $(-\infty, 10]$ and increasing in $[10, \infty)$
$f(x)$ is increasing in $(-\infty, 10]$ and decreasing in $[10, \infty)$
$\mathrm{f}(x)$ is increasing throughout real line
$\mathrm{f}(x)$ is decreasing throughout real line

Solution:

Concept:

A differentiable function is increasing when its derivative is positive for all real values of

x

Explanation:

Given

f(x)=x^{3}-10x^{2}+200x-10

, differentiate with respect to

x

f'(x)=3x^{2}-20x+200

Now check whether

f'(x)

is always positive.

f'(x)=3\left[\left(x-\frac{10}{3}\right)^{2}+\frac{500}{9}\right]

Since

\left(x-\frac{10}{3}\right)^{2} \ge 0

for every real

x

f'(x) \ge \frac{500}{3} > 0

f'(x)>0

for all real

x

, meaning

f(x)

is increasing throughout the real line.

Answer:

Option C:

f(x)

is increasing throughout the real line.

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