Question

# Find the points of the local maxima or local minima and corresponding local maximum and local minimum values of the following function. Also, find the points of inflection, if any.$$f(x)=x^3-6x^2+9x+15$$

Solution

## $$f(x) = x^3 - 6x^2 + 9x + 15$$Differentiate w.r.t $$x$$, we get,$$f'(x) = 3x^2 - 12x +9$$$$3 (x^2 - 4x + 3)$$$$= 3 (x - 3)(x-1)$$For critical points $$f'(x) = 0$$$$\Rightarrow 3(x-3) (x-1) = 0$$$$\Rightarrow x = 3$$At $$x = 1$$, $$f'(x)$$ changes from $$+Ve$$ to $$-Ve$$,hence, $$x = 1$$ is point of maximum $$\boxed{f(1) = 19}$$At $$x = 3, f'(x)$$ changes from $$-Ve$$ to $$+Ve$$,hence, $$x = 3$$ is point of maximum $$\boxed{f(3) = 15}$$Mathematics

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