If the function y=g(x) representing the slope | vedclass

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

Question

(D) The slope of the tangent to the curve $y=3x^4-5x^3-12x^2+18x+3$ is given by the derivative $g(x) = \frac{dy}{dx}$.Calculating the derivative: $g(x

Vedclass · Accepted Answer

$R-[-\frac{1}{2}, \frac{4}{3}]$

Vedclass · Answer

(D) The slope of the tangent to the curve $y=3x^4-5x^3-12x^2+18x+3$ is given by the derivative $g(x) = \frac{dy}{dx}$.Calculating the derivative: $g(x) = 12x^3 - 15x^2 - 24x + 18$.For $g(x)$ to be strictly increasing,its derivative $g'(x)$ must be greater than $0$.$g'(x) = 36x^2 - 30x - 24$.Setting $g'(x) > 0$: $36x^2 - 30x - 24 > 0$.Dividing by $6$: $6x^2 - 5x - 4 > 0$.Factoring the quadratic: $(3x-4)(2x+1) > 0$.The roots are $x = \frac{4}{3}$ and $x = -\frac{1}{2}$.The inequality holds for $x \in (-\infty, -\frac{1}{2}) \cup (\frac{4}{3}, \infty)$.This is equivalent to $R - [-\frac{1}{2}, \frac{4}{3}]$.Thus,the correct option is $D$.

If the function $y=g(x)$ representing the slopes of the tangents drawn to the curve $y=3x^4-5x^3-12x^2+18x+3$ is strictly increasing,then the domain of $g(x)$ is:

Similar Questions

Find the intervals in which the function $f(x) = x^{2} + 2x - 5$ is strictly increasing or strictly decreasing.

Let $f(x) = \sin^4 x + \cos^4 x$. Then $f$ is an increasing function in the interval:

In the interval $[0, 1]$,the function $f(x) = x^2 - x + 1$ is

The function $f(x) = 1 - e^{-\frac{x^2}{2}}$ is .......

For what values of $a$ is the function $f(x) = x^{2} + ax + 1$ increasing on the interval $[1, 2]$?

Vedclass Test Series

Exam Paper Generator

Online Exam Module