Question

Let F(x) = f(x)g(x)h(x) for all real x, where f(x), g(x) and h(x) are differentiable functions. At some point ${{\text{x}}_0}$, ${\text{F' (}}{{\text{x}}_0}){\text{ = 21 F(}}{{\text{x}}_0})$, ${\text{f' (}}{{\text{x}}_0}){\text{ = 4 f(}}{{\text{x}}_0})$, ${\text{g' (}}{{\text{x}}_0}){\text{ = - 7 g(}}{{\text{x}}_0})$ and ${\text{h' (}}{{\text{x}}_0}){\text{ = k h(}}{{\text{x}}_0})$. Then k is equal to

Answer 1

Hint: In this question we will apply the product rule of differentiation and also use the given conditions to find the value of k to solve the given question. We will apply the differentiation on the condition F(x) = f(x)g(x)h(x) and use the other given conditions accordingly.

Complete step-by-step answer:

Now, we are given F(x) = f(x)g(x)h(x). Also, there are conditions given in the question. ${\text{F' (}}{{\text{x}}_0}){\text{ = 21 F(}}{{\text{x}}_0})$, ${\text{f' (}}{{\text{x}}_0}){\text{ = 4 f(}}{{\text{x}}_0})$, ${\text{g' (}}{{\text{x}}_0}){\text{ = - 7 g(}}{{\text{x}}_0})$, ${\text{h' (}}{{\text{x}}_0}){\text{ = k h(}}{{\text{x}}_0})$.

Now by seeing the condition we know that we have to use differentiation. We will use the product rule of differentiation in F(x) = f(x)g(x)h(x). Now, product rule states that if a function h(x) is the product of function f(x) and g(x) and derivative of h(x) is written as,
h(x) = f(x) g(x)
$\dfrac{{{\text{d h(x)}}}}{{{\text{dx}}}}{\text{ = g(x)}}\dfrac{{{\text{d f(x)}}}}{{{\text{dx}}}}{\text{ + f(x)}}\dfrac{{{\text{d g(x)}}}}{{{\text{dx}}}}$

Now, differentiating F(x) = f(x)g(x)h(x) both sides with respect to x and applying product rule, we get
${\text{F' (x) = f'(x)g(x)h(x) + f(x)g'(x)h(x) + f(x)g(x)h'(x)}}$

Now, using conditions given in the questions in the above equation, we get
21 F(x) = 4f(x)g(x)h(x) – 7f(x)g(x)h(x) + kf(x)g(x)h(x)

Putting value of F(x) in the above equation,
$\Rightarrow$ 21(f(x)g(x)h(x)) = 4f(x)g(x)h(x) – 7f(x)g(x)h(x) + kf(x)g(x)h(x)

Taking f(x)g(x)h(x) common from both sides,
$\Rightarrow$ 21(f(x)g(x)h(x)) = f(x)g(x)h(x) (4 – 7 + k)
$\Rightarrow$ 21 = 4 – 7 + k
$\Rightarrow$ k = 24

So, the value of k is 24.

Note: To solve such types of questions we will follow a few steps to find the solution of the given problem. First, we will select the condition with which we will start. Then, we will apply the differentiation on that property. After it, we will use the given conditions in the question to solve the question correctly.