Maximum slope of the curve $y = -x^3 + 3x^2 + 9x - 27$ is

$y = -x^3 + 3x^2 + 9x -27$ Slope $= \frac{dy}{dx} = m = - 3 x^{ 2} + 6 x + 9$ Now, $\frac{dm}{dx} = - 6x + 6$ Now, $\frac{dm}{dx} = 0$ $\Rightarrow -6x + 6 = 0$ $\Rightarrow x = 1$ Now, $\frac{d^{2}m}{dx^{2}} =-6 < 0 \,\forall\, x$ $\therefore x = 1$ is a point of local maximum. $\therefore$ Maximum slope $= -3\left(1\right)^{2 }+ 6\left(1\right) + 9 = 12$

Maximum slope of the curve $y = -x^3 + 3x^2 + 9x -

Notes on Application of derivatives

Concepts Used:

Application of Derivatives

Various Applications of Derivatives-

Rate of Change of Quantities:

If some other quantity ‘y’ causes some change in a quantity of surely ‘x’, in view of the fact that an equation of the form y = f(x) gets consistently pleased, i.e, ‘y’ is a function of ‘x’ then the rate of change of ‘y’ related to ‘x’ is to be given by

$\frac{\triangle y}{\triangle x}=\frac{y_2-y_1}{x_2-x_1}$

This is also known to be as the Average Rate of Change.

Increasing and Decreasing Function:

Consider y = f(x) be a differentiable function (whose derivative exists at all points in the domain) in an interval x = (a,b).

If for any two points x₁ and x₂ in the interval x such a manner that x₁ < x₂, there holds an inequality f(x₁) ≤ f(x₂); then the function f(x) is known as increasing in this interval.
Likewise, if for any two points x₁ and x₂ in the interval x such a manner that x₁ < x₂, there holds an inequality f(x₁) ≥ f(x₂); then the function f(x) is known as decreasing in this interval.
The functions are commonly known as strictly increasing or decreasing functions, given the inequalities are strict: f(x₁) < f(x₂) for strictly increasing and f(x₁) > f(x₂) for strictly decreasing.

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