Q.

Let a∈R and f:R→R be given by f(x)=x5−5x+a then,

see full answer

Start JEE / NEET / Foundation preparation at rupees 99/day !!

21% of IItians & 23% of AIIMS delhi doctors are from Sri Chaitanya institute !!
An Intiative by Sri Chaitanya

a

f(x) has three real roots, if a > 4

b

f(x) has only one real root, if a > 4

c

f(x) has three real roots, if a < - 4

d

f(x) has three real roots, if - 4

answer is B.

(Unlock A.I Detailed Solution for FREE)

Ready to Test Your Skills?

Check your Performance Today with our Free Mock Test used by Toppers!

Take Free Test

Detailed Solution

Let y=x5−5x  (i) As x→∞,y→∞ and x→−∞,y→−∞(ii) Also, at x = 0, y = 0, thus the curve passes through the origin. (iii) dydx=5x4−5=5x4−1=5x2−1x2+1        =5(x−1)(x+1)x2+1Now, dydx>0 in (−∞,−1)∪(1,∞) thus f(x) is increasing in these intervalAlso, dydx<0(−1,1) thus decreasing in (-1, 1).(iv) Also , at x = -1,  dydx changes its sign from + ve to vex= - 1 is point of local maxima.Similarly, x = 1 is point of local minimaLocal maximum value,y=(−1)5−5(−1)=4Local minimum value,y=(1)5−5(1)=−4Now, let y = -aAs, evident from the graph, if −a∈(−4,4) i.e., a∈(−4,4)Then, f(x) has three real roots and if −a>4 or −a>−4 then f(x) has one real root.
Watch 3-min video & get full concept clarity
score_test_img

Get Expert Academic Guidance – Connect with a Counselor Today!

whats app icon
Let a∈R and f:R→R be given by f(x)=x5−5x+a then,