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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 $$\boxed{f(3) = 15}$$

1494259_1661162_ans_a17ed02954f546d7ab18dd48ab3c248e.png

Mathematics

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