The number of distinct real roots of x^4-4x^3 | vedclass

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

Question

(A) Let $f(x) = x^4 - 4x^3 + 12x^2 + x - 1$.First,find the derivative $f'(x) = 4x^3 - 12x^2 + 24x + 1$.Now,find the second derivative $f''(x) = 12x^2

Vedclass · Accepted Answer

$2$

Vedclass · Answer

(A) Let $f(x) = x^4 - 4x^3 + 12x^2 + x - 1$.First,find the derivative $f'(x) = 4x^3 - 12x^2 + 24x + 1$.Now,find the second derivative $f''(x) = 12x^2 - 24x + 24 = 12(x^2 - 2x + 2)$.The discriminant of the quadratic $x^2 - 2x + 2$ is $D = (-2)^2 - 4(1)(2) = 4 - 8 = -4 Since $D Thus,$f'(x) = 0$ has exactly one real root.Since $f'(x)$ has only one real root,$f(x)$ can have at most two real roots.Evaluating $f(x)$ at some points: $f(0) = -1$ and $f(1) = 9$.Since the sign changes between $x=0$ and $x=1$,there is at least one root in $(0, 1)$.Also,$f(-1) = 1 + 4 + 12 - 1 - 1 = 15 > 0$,so there is at least one root in $(-1, 0)$.Therefore,the equation has exactly $2$ distinct real roots.

The number of distinct real roots of $x^4-4x^3+12x^2+x-1=0$ is

Similar Questions

Divide $20$ into two parts such that the product of one part and the cube of the other is maximum. The two parts are

If $PQ$ and $PR$ are the two sides of a triangle,then the angle between them which gives the maximum area of the triangle is

Two stones are thrown vertically upwards with equations of motion $s_1 = 19.6t - 4.9t^2$ and $s_2 = 9.8t - 4.9t^2$ respectively. The maximum height of the first stone is $h$. When the first stone is at its maximum height,the height of the second stone is:

The maximum volume (in cubic units) of the cylinder which can be inscribed in a sphere of diameter $6$ units is

Find the absolute maximum value and the absolute minimum value of the function given by $f(x) = \sin x + \cos x$ for $x \in [0, \pi]$.

Vedclass Test Series

Exam Paper Generator

Online Exam Module