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Question

Investigate the behaviour of the function y=(x3+4)(x+1)3 and construct its graph. How many solutions does the equation (x3+4)(x+1)3=c possess ?

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Solution


y=(x3+4)(x+1)3 ...... (1)
point of intersection on axis
At x=0,y=4×(1)2=4
At y=0,(x3+4)(x+1)3=0
x=1
x=(4)1/3
Increasing and decreasing nature
dydx=(x3+4)×3(x+1)2+(x+1)3(3x2)
dydx=(x+1)2[3x3+12+3x2]
dydx=3(x+1)2[x3+x2+4] ..... (2)
At x=1,2 from above turning point exist
For maxima and minima
y=[(1)3+4](1+1)3,y=0
y=[(2)3+4](2+1)3,y=4
(Image)
(x3+4)(x+1)3=C
So, 2 solutions exist.

887069_262405_ans_d7d5bbf663354b018e251909eb6ff1f2.JPG

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