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Question: If \[f(x)={{x}^{5}}-20{{x}^{3}}+240x,\] then \( f(x) \) satisfies which of the following A. It is ...

If f(x)=x520x3+240x,f(x)={{x}^{5}}-20{{x}^{3}}+240x, then f(x)f(x) satisfies which of the following
A. It is monotonically decreasing everywhere
B. It is monotonically decreasing only in (0,)\left( 0,\infty \right)
C. It is monotonically increasing everywhere
D. It is monotonically increasing only in (,0)\left( -\infty ,0 \right)

Explanation

Solution

We will use the concept of differentiation to find whether it is monotonically increasing or decreasing and in what range. According to the concept, if the first derivative of a function gives zero then there will be an interval for increasing and decreasing and if the first derivative is not zero then the function may be either monotonically increasing or decreasing.

Complete step by step answer:
Moving ahead with the question in step wise manner;
We are asked to find whether the function is monotonically increasing or decreasing and in what range. Since according to the concept of differentiation we know that if the first derivative of function gives zero then at that point function will change its character from increasing to decreasing or decreasing to increasing, i.e. if the function will be decreasing before the point it give its first derivative zero then after that point it will be continuously increasing till the next point comes where the first derivative is zero.
And if the first derivative of a function does not come zero then the function is either continuously increasing or continuously increasing everywhere. To find whether function is continuously increasing or decreasing we can come to point by seeing its character i.e. for a function to be increasing the function at some point ‘a’ should be greater than some point ‘b’ if point b>ab>a , as it can be denotes as for continuously increasing function f(b)>f(a);b>af(b)>f(a);b>a .
Now for our case let us first find the points at which it gives zero. For that let us first find the first derivative, i.e. df(x)dx\dfrac{df(x)}{dx} , which will be equal to;

& \dfrac{df(x)}{dx}=\dfrac{d\left( {{x}^{5}}-20{{x}^{3}}+240x \right)}{dx} \\\ & f'(x)=5{{x}^{4}}-60{{x}^{2}}+240 \\\ \end{aligned}$$ Now to find the zeros of the first derivative, put it equal to zero and find its roots. So we will get; $$5{{x}^{4}}-60{{x}^{2}}+240=0$$ To find the roots of such biquadratic let us assume $ {{x}^{4}}={{X}^{2}} $ then we will get the equation; $$5{{X}^{2}}-60X+240=0$$. So now it seems to be like a quadratic whose roots we can find easily. As we know that if the ‘D’ i.e. $ {{b}^{2}}-4ac $ of a quadratic equation is less than zero then there are no real roots of the quadratic equation. So we can say that there are no roots of the quadratic equation$$5{{x}^{4}}-60{{x}^{2}}+240=0$$. Which means there is no value which will give zero for the first derivative of function. So by the concept of differentiation if the first derivative does not give zero then the function is continuously increasing or decreasing everywhere. Hence we can say that the function$$f(x)={{x}^{5}}-20{{x}^{3}}+240x$$ is either continuously increasing or continuously decreasing. Now to find whether it is increasing or decreasing let us check whether it follows the rule i.e. $ f(b)>f(a);b>a $ .if it does not follow then the function is decreasing and if follows then the function is increasing. So let the value of ‘a’ be equal to $ -1 $ and that of ‘b’ is $ 1 $ i.e. $ a=-1.b=1 $ as it satisfies that $ b>a $ . So putting the value of ‘a’ and ‘b’ in the condition $ f(b)>f(a);b>a $ , we will get $ f(a) $ ; $$\begin{aligned} & f(x)={{x}^{5}}-20{{x}^{3}}+240x \\\ & f(-1)={{\left( -1 \right)}^{5}}-20{{\left( -1 \right)}^{3}}+240\left( -1 \right) \\\ & f(-1)=-1+20-240 \\\ & f(-1)=-221 \\\ \end{aligned}$$ And $ f(b) $ is equal to: $$\begin{aligned} & f(x)={{x}^{5}}-20{{x}^{3}}+240x \\\ & f(1)={{\left( 1 \right)}^{5}}-20{{\left( 1 \right)}^{3}}+240\left( 1 \right) \\\ & f(1)=1-20+240 \\\ & f(1)=221 \\\ \end{aligned}$$ So we got $ f(a) $ equal to $ -221 $ and $ f(b) $ equal to $ 221 $ , which means $ f(b)>f(a) $ . Hence it satisfies the condition $ f(b)>f(a);b>a $ . Means the function is continuously increasing everywhere. Hence the function is monotonically increasing everywhere. **So, the correct answer is “Option C”.** **Note:** Monotonically means continuously, i.e. monotonically increasing or decreasing means continuously increasing or decreasing. Moreover if we get the roots from first derivative then by using the condition $ f(b)>f(a);b>a $ only we will be finding whether it is increasing or decreasing and point ‘a’ and ‘b’ will be the points present on either sides of points that give first derivative of function zero.