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In how many points graph of y=x^3-3x^2+5...

In how many points graph of `y=x^3-3x^2+5x-3` intersect the x-axis?

Text Solution

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The number of points in which ` y = x^(3) - 3x^(2) + 5x - 3` intersects the x-axis is same as the number of real roots of the equation ` x ^(3) - 3x^(2) + 5x - 3 = 0`.
Now we can see that x = 1 satisfies the equation, hence one root of the equation is x = 1.
Now dividing ` x^(3) - 3x^(2) + 5x-3` by x - 1 , we have the quotient ` x^(2) - 2x + 3`.
Hence the equation reduces to `(x-1)(x^(2)-2x+3)= 0`.
Hence the only root of the equation is x = 1.
Thus, the graph of ` y = x^(3) - 3x^(2) + 5x - 3` cuts the x - axis in one point only.
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Following is the graph of y = f' (x) , given that f(c) = 0. Analyse the graph and answer the following questions. (a) How many times the graph of y = f(x) will intersect the x - axis? (b) Discuss the type of roots of the equation f (x) = 0, a le x le b . (c) How many points of inflection the graph of y = f(x), a le x le b , has? ,(d) Find the points of local maxima/minima of y = f(x), a le x le b , , (e) f"(x)=0 has how many roots?

Knowledge Check

  • The straight line 2x-3y+5=0 intersects the x -axis and y- axis-

    A
    `((5)/(2),0)and(0,(5)/(3))` respectively
    B
    `(-(5)/(2),0)and(0,-(5)/(3))` respectively
    C
    `((5)/(2),0)and(0,-(5)/(3))` respectively
    D
    `(-(5)/(2),0)and(0,(5)/(3))` respectively

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