CBSE (Commerce) Class 12CBSE
Share
Notifications

View all notifications
Books Shortlist
Your shortlist is empty

Solution for Find the Interval in Which the Following Function Are Increasing Or Decreasing F(X) = X3 − 6x2 − 36x + 2 ? - CBSE (Commerce) Class 12 - Mathematics

Login
Create free account


      Forgot password?

Question

Find the interval in which the following function are increasing or decreasing f(x) = x3 − 6x2 − 36x + 2 ?

Solution

\[\text { When } \left( x - a \right)\left( x - b \right)>0 \text { with }a < b, x < a \text { or }x>b.\]

\[\text { When } \left( x - a \right)\left( x - b \right)<0 \text { with } a < b, a < x < b .\]

\[f\left( x \right) = x^3 - 6 x^2 - 36x + 2\]

\[f'\left( x \right) = 3 x^2 - 12x - 36\]

\[ = 3 \left( x^2 - 4x - 12 \right)\]

\[ = 3 \left( x - 6 \right)\left( x + 2 \right)\]

\[\text { For }f(x) \text { to be increasing, we must have },\]

\[f'\left( x \right) > 0\]

\[ \Rightarrow 3 \left( x - 6 \right)\left( x + 2 \right) > 0\]

\[ \Rightarrow \left( x - 6 \right)\left( x + 2 \right) > 0 \left[ \text { Since } 3 > 0, 3 \left( x - 6 \right)\left( x + 2 \right) > 0 \Rightarrow \left( x - 6 \right)\left( x + 2 \right) > 0 \right]\]

\[ \Rightarrow x < - 2 \ or \ x > 6\]

\[ \Rightarrow x \in \left( - \infty , - 2 \right) \cup \left( 6, \infty \right)\]

\[\text { So,}f(x)\text { is increasing on } x \in \left( - \infty , - 2 \right) \cup \left( 6, \infty \right).\]

\[\text { For }f(x) \text { to be decreasing, we must have }\]

\[f'\left( x \right) < 0\]

\[ \Rightarrow 3 \left( x - 6 \right)\left( x + 2 \right) < 0\]

\[ \Rightarrow \left( x - 6 \right)\left( x + 2 \right) < 0 \left[ \text { Since } 3 > 0, 3 \left( x - 6 \right)\left( x + 2 \right) < 0 \Rightarrow \left( x - 6 \right)\left( x + 2 \right) < 0 \right]\]

\[ \Rightarrow - 2 < x < 6 \]

\[ \Rightarrow x \in \left( - 2, 6 \right)\]

\[\text{ So },f(x)\text { is decreasing on } x \in \left( - 2, 6 \right) .\]

  Is there an error in this question or solution?
Solution Find the Interval in Which the Following Function Are Increasing Or Decreasing F(X) = X3 − 6x2 − 36x + 2 ? Concept: Increasing and Decreasing Functions.
S
View in app×