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Question: If \( f\left( x \right) = {x^3} + b{x^2} + cx + d \) and \( 0 < {b^2} < c \) , then in \( \left( { -...

If f(x)=x3+bx2+cx+df\left( x \right) = {x^3} + b{x^2} + cx + d and 0<b2<c0 < {b^2} < c , then in (−∞,∞)\left( { - \infty ,\infty } \right)
A. f(x)f\left( x \right) is a strictly increasing function
B. f(x)f\left( x \right) has a local maxima
C. f(x)f\left( x \right) is a strictly decreasing function
D. f(x)f\left( x \right) is bounded

Explanation

Solution

Before solving the question, we should know what maxima and minima of a function mean. Maxima and minima of any function are collectively known as extrema. Extrema are just the maximum and minimum values of a function respectively. The local maxima is the maximum value of the function in a given range while local minima is the minimum value of the function in a given range. To find the local extrema, we will have to find the critical points of the function f(x)f\left( x \right) by differentiating it and then equating it to zero. Here, in this question we will first have to differentiate the given function and then proceed.

Complete step by step solution:
Given function is f(x)=x3+bx2+cx+df\left( x \right) = {x^3} + b{x^2} + cx + d
First, we differentiate the given function and get,
f′(x)=3x2+2bx+cf'\left( x \right) = 3{x^2} + 2bx + c
As we very well know that ax2+bx+c>0a{x^2} + bx + c > 0 for all x⇒a>0x \Rightarrow a > 0 and D<0D < 0 in the above equation. So,
D=4b2−12c D=4(b2−c)−8c  D = 4{b^2} - 12c \\\ D = 4\left( {{b^2} - c} \right) - 8c \\\
Where, b2−c<0{b^2} - c < 0 and c>0c > 0
D=(−ve)  or  D<0D = \left( { - ve} \right)\;or\;D < 0
Therefore,
f′(x)=3x2+2bx+c>0f'\left( x \right) = 3{x^2} + 2bx + c > 0 for all x∈(−∞,∞)x \in \left( { - \infty ,\infty } \right) as D<0D < 0 and a>0a > 0
So, f(x)=x3+bx2+cx+df\left( x \right) = {x^3} + b{x^2} + cx + d is a strictly increasing function.

So, the correct answer is Option A.

Note: In order to solve these types of questions, we should have the basic knowledge of trigonometric functions. Along with that, we should also know about maxima and minima. Students can find many similar questions in the NCERT book to start with and then they can even proceed to competitive books to enhance their knowledge. Students should remember that the local maximum value or local minimum value is not the same as relative maximum value or the relative minimum value.