If the set of all values of a,for which the e | vedclass

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

Question

(D) Let $f(x) = 5x^3 - 15x$. The equation $5x^3 - 15x - a = 0$ can be written as $f(x) = a$.To have three distinct real roots,the horizontal line $y =

Vedclass · Accepted Answer

$30$

Vedclass · Answer

(D) Let $f(x) = 5x^3 - 15x$. The equation $5x^3 - 15x - a = 0$ can be written as $f(x) = a$.To have three distinct real roots,the horizontal line $y = a$ must intersect the graph of $f(x)$ at three distinct points.First,find the critical points of $f(x)$ by setting $f'(x) = 0$:$f'(x) = 15x^2 - 15 = 15(x^2 - 1) = 15(x - 1)(x + 1)$.The critical points are $x = 1$ and $x = -1$.The local maximum value is $f(-1) = 5(-1)^3 - 15(-1) = -5 + 15 = 10$.The local minimum value is $f(1) = 5(1)^3 - 15(1) = 5 - 15 = -10$.For the equation $f(x) = a$ to have three distinct real roots,$a$ must lie strictly between the local minimum and local maximum values:$-10 Thus,the interval $(\alpha, \beta)$ is $(-10, 10)$,so $\alpha = -10$ and $\beta = 10$.We need to calculate $\beta - 2\alpha$:$\beta - 2\alpha = 10 - 2(-10) = 10 + 20 = 30$.

If the set of all values of $a$,for which the equation $5x^3 - 15x - a = 0$ has three distinct real roots,is the interval $(\alpha, \beta)$,then $\beta - 2\alpha$ is equal to . . . . . . .

Similar Questions

The minimum value of $\left( x^2 + \frac{250}{x} \right)$ is

At what points in the interval $[0, 2\pi]$,does the function $\sin 2x$ attain its maximum value?

For all real $x$,the minimum value of $\frac{1-x+x^{2}}{1+x+x^{2}}$ is

If the function $y = a \log x + bx^2 + x$ has extreme values at $x = 1$ and $x = 2$,then $(a, b) = \dots$

The height of the cone of maximum volume inscribed in a sphere of radius $R$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module