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What are all values of k for which the graph y=x

3

−3x

2

+k will have three distinct x-intercepts?

Asked by supergirl3210 • 05/07/2023

0:20

/

0:38

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Values of k for which the **graph **y = x³ - 3x² + k will have three distinct x - intercepts are given by 0 < k < 4.

Given the **function **of the **graph **is,

y = x³ - 3x² + k, where k is constant.

**Differentiating **the function with respect to 'x' we get,

y' = 3x² - 6x

y'' = 6x - 6

Now, y' = 0 gives

3x² - 6x = 0

x² - 2x = 0

x(x - 2) = 0

x = 0, 2

Now, for x = 0, y''(0) = -6 < 0, so at this point the function has maximum value.

and for x = 2, y''(2) = 12 - 6 = 6 > 0, so at this point the function has minimum value.

So, max y = y(0) = k > 0

min y = y(2) = 8 - 12 + k = k - 4 < 0

So, k < 4

Hence the values are, 0 < k < 4.