Consider the function `h(x)=(g^(2)(x))/(2)+3x^(3)-5`, where g(x) is a continuous and differentiable function. It is given that h(x) is a monotonically increasing function and g(0) = 4. Then which of the following is not true ?

Options

A. `g^(2)(1)gt10`
B. `h(5)gt3`
C. `h((5)/(2))lt2`
D. `g^(-1)lt22`

Explanation

To solve the problem, we need to analyze the function \( h(x) = \frac{g^2(x)}{2} + 3x^3 - 5 \) and determine which statement about \( h(x) \) is not true, given that \( h(x) \) is a monotonically increasing function and \( g(0) = 4 \).

### Step 1: Understand the condition of monotonicity
Since \( h(x) \) is a monotonically increasing function, its derivative \( h'(x) \) must be greater than or equal to 0 for all \( x \).

### Step 2: Calculate the derivative of \( h(x) \)
The derivative of \( h(x) \) is given by:
\[ h'(x) = \frac{d}{dx}\left(\frac{g^2(x)}{2}\right) + \frac{d}{dx}(3x^3) - \frac{d}{dx}(5) \]
Using the chain rule, we have:
\[ h'(x) = g(x)g'(x) + 9x^2 \]

### Step 3: Set up the inequality for monotonicity
Since \( h'(x) \geq 0 \), we can write:
\[ g(x)g'(x) + 9x^2 \geq 0 \]

### Step 4: Analyze the inequality
This implies:
\[ g(x)g'(x) \geq -9x^2 \]
This means that the product \( g(x)g'(x) \) must be greater than or equal to \(-9x^2\).

### Step 5: Evaluate at \( x = 0 \)
Given \( g(0) = 4 \):
\[ g(0)g'(0) \geq -9(0)^2 \implies 4g'(0) \geq 0 \]
This implies \( g'(0) \geq 0 \).

### Step 6: Evaluate at \( x = 1 \)
Now, let's evaluate the inequality at \( x = 1 \):
\[ g(1)g'(1) + 9(1)^2 \geq 0 \implies g(1)g'(1) + 9 \geq 0 \]
This means \( g(1)g'(1) \geq -9 \).

### Step 7: Evaluate \( h(0) \) and \( h(5) \)
Now, let's find \( h(0) \) and \( h(5) \):
\[ h(0) = \frac{g^2(0)}{2} + 3(0)^3 - 5 = \frac{4^2}{2} - 5 = \frac{16}{2} - 5 = 8 - 5 = 3 \]
\[ h(5) = \frac{g^2(5)}{2} + 3(5)^3 - 5 = \frac{g^2(5)}{2} + 375 - 5 = \frac{g^2(5)}{2} + 370 \]

### Step 8: Determine which statement is not true
Since \( h(x) \) is increasing, we have \( h(5) > h(0) \), which implies:
\[ \frac{g^2(5)}{2} + 370 > 3 \implies \frac{g^2(5)}{2} > -367 \]
This is always true since \( g^2(5) \geq 0 \).

### Conclusion
Given the analysis, we can conclude that the statement which is not true must be one that contradicts the monotonicity of \( h(x) \) or the properties derived from \( g(x) \).