Question

If $ y = {x^3} - a{x^2} + 48x + 7 $ is an increasing function of all real values of $ x $ , then $ a $ lies in:
a) $ ( - 14,14) $
b) $ ( - 12,12) $
c) $ ( - 16,16) $
d) $ ( - 21,21) $

Answer 1

Hint: The given question is from derivatives. We will differentiate the given function once and then we will obtain a quadratic function. A quadratic function is always decreasing if its discriminant is negative. We will use this fact and find the interval where this happens. After doing this we will get the required answer.
Formula used:
1)The discriminant of a quadratic function $ f(x) = A{x^2} + Bx + C $ is given by the formula:
 $ D = {B^2} - 4AC $
2) $ (a + b)(a - b) = {a^2} - {b^2} $

Complete step-by-step answer:
The given equation is $ y = {x^3} - a{x^2} + 48x + 7 $ .
Differentiating the above equation with respect to the variable $ 'x' $ we have:
 $ y' = 3{x^2} - 2ax + 48 $
The above function we got is a quadratic function.
For the above function to be an increasing one for all values of $ 'x' $ ,
Discriminant of the above quadratic function should be less than zero.
 $ D < 0 $
In $ y' = 3{x^2} - 2ax + 48 $ we have $ A = 3,B = - 2a,C = 48 $ so using this in the discriminant formula we get:
 $ \Rightarrow {( - 2a)^2} - 4 \cdot 3 \cdot 48 < 0 $
 $ \Rightarrow 4{a^2} - 4 \cdot 3 \cdot 48 < 0 $
 $ \Rightarrow 4({a^2} - 3 \cdot 48) < 0 $
 $ \Rightarrow {a^2} - 3 \cdot 48 < 0 $
 $ \Rightarrow {a^2} - 144 < 0 $
\[ \Rightarrow {a^2} - {12^2} < 0\]
\[ \Rightarrow (a - 12)(a + 12) < 0\]
\[ \Rightarrow a \in ( - 12,12)\]
Our final answer corresponds to option ‘b’
So, the correct answer is “Option B”.

Note: The most important tool to use in this problem is to use derivative and find the value where the discriminant is negative. Take care of the inequality, it is always strictly less than zero not less than or equal to zero.